Discrete Math Quiz: Argument Validity
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Argument ValidityQuestion 1 of 20

Evaluate this mathematical argument: "Suppose x2=4x^2 = 4. Then either x=2x = 2 or x=−2x = -2. Since we know x>0x > 0, we can conclude x=2x = 2." Which logical principle does this argument primarily employ?

Disjunctive syllogism combined with additional constraint information to eliminate one disjunct
Modus tollens applied to the conditional relationship between sign and value
Proof by cases where each case is systematically verified independently
Reductio ad absurdum by assuming the opposite of the desired conclusion
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Discrete Math Quiz

Discrete Math Quiz: Argument Validity

Practice Argument Validity in Discrete Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Argument Validity, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Evaluate this mathematical argument: "Suppose x2=4x^2 = 4. Then either x=2x = 2 or x=−2x = -2. Since we know x>0x > 0, we can conclude x=2x = 2." Which logical principle does this argument primarily employ?

  1. Disjunctive syllogism combined with additional constraint information to eliminate one disjunct (correct answer)
  2. Modus tollens applied to the conditional relationship between sign and value
  3. Proof by cases where each case is systematically verified independently
  4. Reductio ad absurdum by assuming the opposite of the desired conclusion
Explanation: The argument establishes a disjunction (x = 2 or x = -2) from the equation, then uses the additional constraint (x > 0) to eliminate one disjunct (x = -2), leaving x = 2. This is disjunctive syllogism: P ∨ Q, ¬Q, therefore P. Choice B is wrong because we're not using modus tollens. Choice C is wrong because we're not verifying cases independently. Choice D is wrong because we're not using proof by contradiction.

Question 2

A student argues: "Every recursive function that terminates has a base case. This function has a base case. Therefore, this recursive function terminates." Another student counters: "That's like saying every bird that flies has wings, penguins have wings, so penguins fly." What does the analogy reveal about the original argument?

  1. Both arguments are valid but demonstrate different logical principles in computer science versus biology
  2. Both arguments commit the same logical fallacy of affirming the consequent (correct answer)
  3. The analogy is flawed because recursive functions and birds follow different logical rules
  4. The original argument is valid, but the analogy incorrectly suggests it's fallacious
Explanation: Both arguments have the form: All A that are B have property C. X has property C. Therefore, X is A that is B. This is affirming the consequent (if P then Q, Q, therefore P). Having a base case is necessary but not sufficient for termination - the function might still recurse infinitely without reaching the base case. Choice A is wrong because both are invalid. Choice C is wrong because the logical structure is the same. Choice D is wrong because the original argument is fallacious.

Question 3

Analyze this proof by contradiction setup: "Assume for contradiction that there exists a largest prime number P. Consider the number N = P! + 1, where P! is the factorial of P. Since N > P, if N is prime, then N is larger than the supposedly largest prime P." What logical step is needed to complete this contradiction?

  1. Show that N must be prime by proving it's not divisible by any number less than P
  2. Show that N has a prime factor greater than P, contradicting P being the largest prime (correct answer)
  3. Show that N is composite, leading to a contradiction with our construction method
  4. Show that P! + 1 creates an infinite sequence, contradicting finiteness of primes
Explanation: The key insight is that N = P! + 1 is not divisible by any prime ≤ P (since dividing by any such prime leaves remainder 1). Therefore, N either is prime itself (and > P) or has a prime factor > P. Either way, there exists a prime > P, contradicting P being largest. Choice A is incomplete - we don't need N to be prime. Choice C goes in the wrong direction. Choice D misunderstands the argument structure.

Question 4

Consider the argument: "If the network is secure, then unauthorized access is impossible. The network logs show unauthorized access occurred. Therefore, the network is not secure." Which logical form best represents this argument, and is it valid?

  1. Modus tollens: P→Q,¬Q∴¬PP \rightarrow Q, \neg Q \therefore \neg P; the argument is valid (correct answer)
  2. Modus ponens: P→Q,P∴QP \rightarrow Q, P \therefore Q; the argument is valid
  3. Affirming the consequent: P→Q,Q∴PP \rightarrow Q, Q \therefore P; the argument is invalid
  4. Denying the antecedent: P→Q,¬P∴¬QP \rightarrow Q, \neg P \therefore \neg Q; the argument is invalid
Explanation: This argument follows modus tollens: If P then Q, not Q, therefore not P. Here P = "network is secure", Q = "unauthorized access is impossible". We have the conditional P→Q, we observe ¬Q (unauthorized access occurred), so we can validly conclude ¬P (network is not secure). Choice B is wrong because this isn't affirming the antecedent. Choice C is wrong because we're not affirming the consequent. Choice D is wrong because we're not denying the antecedent.

Question 5

A computer science student argues: "In any sorting algorithm, if the input is already sorted, then the algorithm performs optimally. Quicksort performs optimally on this input. Therefore, this input must be already sorted." What logical issue does this argument have?

  1. It contains no logical errors but makes an empirically false claim about quicksort
  2. It commits denying the antecedent - the premise about sorting algorithms is too broad
  3. It uses faulty universal generalization - not all algorithms behave the same way
  4. It commits affirming the consequent - optimal performance can occur for reasons other than pre-sorted input (correct answer)
Explanation: When analyzing logical arguments, you need to identify the structure and check whether the conclusion follows validly from the premises. This argument follows the pattern: "If P, then Q. Q is true. Therefore, P is true." The student's reasoning commits the logical fallacy of affirming the consequent. Just because quicksort performs optimally AND the premise states that algorithms perform optimally on sorted input, you cannot conclude the input was sorted. Optimal performance can occur for other reasons - perhaps the input was reverse-sorted, or had some other special property that made quicksort efficient. The premise only tells us that sorted input guarantees optimal performance, not that optimal performance only comes from sorted input. Option A is wrong because this isn't about empirical claims regarding quicksort's behavior - it's about logical structure. The argument would be invalid even if all the factual claims were true. Option B incorrectly identifies the fallacy as denying the antecedent, which would involve concluding "not Q" from "not P." Option C misses the mark entirely - this isn't about overgeneralization but about invalid conditional reasoning. The correct answer is D because affirming the consequent assumes that if a condition is sufficient for an outcome, it must also be necessary. But sufficient conditions aren't automatically necessary conditions. Remember: When you see conditional statements in logic problems, always check the direction of the reasoning. "If P then Q" only guarantees Q when P is true - it doesn't work backwards.

Question 6

Consider this argument involving quantifiers: "All computer science students take discrete math. Some students who take discrete math struggle with proofs. Therefore, some computer science students struggle with proofs." What is the validity status of this argument?

  1. Valid, following the pattern of universal-particular syllogism with proper distribution of terms
  2. Invalid, because it commits the fallacy of undistributed middle term in categorical logic
  3. Invalid, because the conclusion goes beyond what the premises logically guarantee (correct answer)
  4. Valid, but only if we assume the existential import of the universal statement
Explanation: This argument is invalid. Let CS = computer science students, DM = students taking discrete math, SP = students struggling with proofs. We have: CS ⊆ DM and DM ∩ SP ≠ ∅. This doesn't guarantee CS ∩ SP ≠ ∅ because the struggling students might be the non-CS students taking discrete math. Choice A is wrong because the middle term issue makes it invalid. Choice B uses correct terminology but the specific fallacy description is imprecise. Choice D is wrong because existential import doesn't resolve the logical gap.

Question 7

Consider the compound argument: "If Alice studies discrete math, then she will pass the exam. If Alice passes the exam, then she will graduate. Alice studies discrete math. Therefore, Alice will graduate." This argument demonstrates which valid inference pattern?

  1. Hypothetical syllogism combined with modus ponens applied to the resulting conditional (correct answer)
  2. Modus tollens applied twice in sequence to reach the final conclusion
  3. Disjunctive syllogism followed by conditional proof of the conclusion
  4. Universal instantiation followed by existential generalization of the conclusion
Explanation: This argument uses hypothetical syllogism to combine the first two conditionals: (P→Q) ∧ (Q→R) gives us P→R. Then modus ponens is applied: we have P→R and P, therefore R. The chain is: studies → passes → graduates, plus "studies", therefore "graduates". Choice B is wrong because modus tollens requires denying the consequent. Choice C is wrong because there's no disjunction involved. Choice D is wrong because this doesn't involve quantifiers.

Question 8

Consider this proof structure: "To prove that P→QP \rightarrow Q, assume P and derive Q. Assume P. [Several steps of reasoning]. Therefore Q. Hence, P→QP \rightarrow Q." A student claims this uses circular reasoning because we assume P to prove P→QP \rightarrow Q. How should this claim be evaluated?

  1. The student is correct - assuming P to prove P→QP \rightarrow Q is indeed circular reasoning
  2. The student is incorrect - this is the standard direct proof method for conditionals (correct answer)
  3. The student is partially correct - this method only works for biconditionals, not simple conditionals
  4. The student is incorrect - this is circular only if Q is used to derive itself within the assumption
Explanation: The student misunderstands direct proof of conditionals. To prove P→Q, we assume P (hypothetically) and show that Q must follow. This isn't circular because we're not assuming what we want to prove (P→Q); we're assuming the antecedent to establish the consequent. Choice A is wrong because this is valid methodology. Choice C is wrong because this works for all conditionals. Choice D is wrong because the issue isn't about using Q to derive Q.

Question 9

A logic student constructs this argument: "No even numbers greater than 2 are prime. All prime numbers are either 2 or odd. 6 is even and greater than 2. Therefore, 6 is not prime." How should this argument be evaluated?

  1. Invalid but sound - the premises are true but the conclusion doesn't follow logically
  2. Valid but unsound - the logical form is correct but the first premise is false
  3. Valid and sound - the conclusion follows logically from true premises (correct answer)
  4. Invalid and unsound - both the logical form and premises have problems
Explanation: When evaluating logical arguments, you need to assess two key components: validity (whether the conclusion follows logically from the premises) and soundness (whether the premises are actually true). Let's examine this argument step by step. The logical structure is: If something is even and greater than 2, then it's not prime. 6 is even and greater than 2. Therefore, 6 is not prime. This follows the valid logical form of modus ponens - the conclusion necessarily follows from the premises. Now check the premises for truth. "No even numbers greater than 2 are prime" - this is mathematically correct. Even numbers are divisible by 2, so any even number greater than 2 has at least three factors (1, 2, and itself), making it composite, not prime. "All prime numbers are either 2 or odd" - also true, since 2 is the only even prime. "6 is even and greater than 2" - obviously true. Since the argument is valid and all premises are true, it's both valid and sound, making C correct. A is wrong because the conclusion does follow logically - the argument is valid. B incorrectly claims the first premise is false when it's actually a fundamental fact about prime numbers. D is wrong because neither the logical form nor the premises have problems. When analyzing logical arguments, always separate validity from soundness. Check if the conclusion follows from the premises (validity), then verify whether the premises themselves are true (soundness). An argument can only be sound if it's first valid.

Question 10

Consider this meta-logical argument: "Every valid argument with true premises has a true conclusion. This argument has true premises and a true conclusion. Therefore, this argument is valid." What is problematic about applying this reasoning to determine argument validity?

  1. It applies proof by contradiction incorrectly to the domain of logical reasoning
  2. It uses circular reasoning by assuming validity to prove validity within the same context
  3. It confuses soundness with validity in the meta-logical analysis of arguments
  4. It commits the fallacy of affirming the consequent when analyzing logical validity (correct answer)
Explanation: When analyzing logical arguments, you need to distinguish between the logical form and the content. This question tests your understanding of conditional reasoning and common logical fallacies in meta-logical analysis. The argument follows this pattern: "If P then Q. Q is true. Therefore P is true." Here, P = "argument is valid" and Q = "argument has true conclusion (given true premises)." This is the classic fallacy of affirming the consequent. Just because a valid argument with true premises yields a true conclusion doesn't mean that every argument with true premises and a true conclusion is valid. The reasoning works only one direction - validity guarantees the conclusion follows from premises, but a true conclusion doesn't guarantee validity. Looking at the wrong answers: A) is incorrect because this isn't attempting proof by contradiction - it's making a direct positive inference. B) misses the mark because the argument isn't assuming validity to prove validity; it's inferring validity from observing true premises and conclusion. C) is wrong because the argument doesn't confuse soundness (valid + true premises) with validity - it actually demonstrates understanding that soundness requires both true premises and validity. The key insight is recognizing conditional statement directionality. When you see "If A then B" followed by "B, therefore A," immediately identify this as affirming the consequent - a fundamental logical fallacy. Remember: necessity flows one way in conditionals, and reversing the direction creates invalid reasoning.

Question 11

Consider this biconditional argument: "A graph is Eulerian if and only if every vertex has even degree. Graph G has all vertices of even degree. Therefore, Graph G is Eulerian." A student objects: "But what if the graph is disconnected?" How does this objection affect the argument's validity?

  1. The argument remains valid because the biconditional was stated correctly and applied properly
  2. The objection shows the argument is invalid because the biconditional statement is incomplete (correct answer)
  3. The argument becomes invalid because disconnected graphs cannot be Eulerian by definition
  4. The objection is irrelevant because connectivity is implied by the even-degree condition
Explanation: The student correctly identifies that the biconditional is incomplete. A graph is Eulerian iff it is connected AND every vertex has even degree. The stated biconditional is missing the connectivity requirement, making the argument invalid as stated. A disconnected graph with all even-degree vertices is not Eulerian. Choice A is wrong because the biconditional was incorrectly stated. Choice C is wrong about why the argument fails. Choice D is wrong because connectivity is not implied by even degrees.

Question 12

Consider the argument: "If the database query is optimized, then response time is under 2 seconds. If response time is under 2 seconds, then user satisfaction is high. The database query is not optimized. Therefore, user satisfaction is not high." What logical fallacy does this argument commit?

  1. False dichotomy - it assumes only two possible states for each variable
  2. Affirming the consequent - the logic incorrectly reverses the conditional relationship
  3. Denying the antecedent - the conclusion doesn't follow from the given premises (correct answer)
  4. Circular reasoning - the conclusion is embedded within the premise structure
Explanation: When analyzing logical arguments, you need to carefully examine the structure of conditional statements and whether the conclusion validly follows from the premises. This question tests your ability to identify common logical fallacies in conditional reasoning. Let's break down the argument structure. We have two conditional statements: "If optimized → response time < 2s" and "If response time < 2s → high satisfaction." The premise states the database is NOT optimized, and concludes satisfaction is NOT high. This commits the fallacy of denying the antecedent. Just because the database isn't optimized doesn't mean response time can't be under 2 seconds through other means, and thus satisfaction could still be high. The conditional "If A then B" doesn't tell us what happens when A is false—B could still be true for other reasons. The conclusion simply doesn't follow logically from the premises. Looking at the wrong answers: (A) False dichotomy would involve incorrectly assuming only two options exist, but this argument isn't about limiting choices. (B) Affirming the consequent would mean concluding the antecedent is true because the consequent is true (like "If A then B, B is true, therefore A is true"), which isn't happening here. (D) Circular reasoning would require the conclusion to be hidden within the premises, but the conclusion about satisfaction is distinct from the premises about optimization and response time. Remember: in conditional logic, denying the antecedent never allows you to deny the consequent. Watch for arguments that make this invalid leap—they're testing whether you understand that conditionals only work in one direction.

Question 13

A student claims: "Either the algorithm terminates or it runs forever. The algorithm doesn't terminate. Therefore, it runs forever." Another student responds: "But what if the algorithm crashes?" Which best describes this logical situation?

  1. The first argument is valid by disjunctive syllogism, but the second student identifies an ambiguity in the premises (correct answer)
  2. The first argument is invalid because it commits the fallacy of false dichotomy
  3. The first argument is valid, and the second student's objection is irrelevant to logical validity
  4. Both students are correct, demonstrating that disjunctive syllogism can have multiple valid conclusions
Explanation: The first argument follows valid disjunctive syllogism: P ∨ Q, ¬P, therefore Q. However, the second student points out that "terminates" might be ambiguous - does crashing count as termination? This highlights that while the logical form is valid, the argument's soundness depends on clarifying the premises. Choice B is wrong because the logical form is valid. Choice C is wrong because ambiguity in premises affects argument quality. Choice D is wrong because disjunctive syllogism has one conclusion given the premises.

Question 14

Analyze this argument about set relationships: "If A⊆BA \subseteq B and B⊆CB \subseteq C, then A⊆CA \subseteq C. We know that A⊈CA \not\subseteq C. Therefore, it's not the case that both A⊆BA \subseteq B and B⊆CB \subseteq C." What logical form and validity does this demonstrate?

  1. Disjunctive syllogism applied to set inclusion relationships; the argument is valid
  2. Denying the antecedent with universal quantification; the argument is invalid
  3. Hypothetical syllogism with negated conclusion; the argument is valid but incomplete
  4. Modus tollens with conjunction in the antecedent; the argument is valid (correct answer)
Explanation: When analyzing logical arguments involving set theory, you need to identify both the logical structure and determine whether the reasoning is valid. This question tests your ability to recognize formal logical patterns applied to mathematical relationships. The argument follows this structure: "If P, then Q. Not Q. Therefore, not P." Here, P represents the conjunction "A⊆BA \subseteq B and B⊆CB \subseteq C" and Q represents "A⊆CA \subseteq C." Since we're given that A⊈CA \not\subseteq C (not Q), we can validly conclude that it's not the case that both subset relationships hold (not P). This is modus tollens, a fundamental valid argument form. The antecedent contains a conjunction, making this modus tollens with conjunction in the antecedent. Option A incorrectly identifies this as disjunctive syllogism, which involves choosing between alternatives ("P or Q, not P, therefore Q"). Option B mischaracterizes this as denying the antecedent, which would be invalid reasoning of the form "If P then Q, not P, therefore not Q." Option C mentions hypothetical syllogism, which chains conditional statements ("If P then Q, if Q then R, therefore if P then R"), but this argument doesn't chain conditionals—it negates the conclusion to deny the premise. Remember that modus tollens is always valid: if you know a conditional statement is true and its conclusion is false, you can definitively conclude the premise is false. Watch for this pattern whenever you see "not the conclusion, therefore not the premise."

Question 15

Examine this argument: "If a graph is planar, then it satisfies Euler's formula V−E+F=2V - E + F = 2. Graph GG has 6 vertices, 9 edges, and 5 faces, so 6−9+5=26 - 9 + 5 = 2. Therefore, GG is planar." What logical issue affects the validity of this reasoning?

  1. Denying the antecedent, since we're reasoning backwards from the formula to the geometric property
  2. The argument is valid because Euler's formula is equivalent to planarity for connected graphs
  3. Invalid calculation, since the face count assumes planarity which creates circular reasoning
  4. Affirming the consequent, since satisfying Euler's formula is necessary but not sufficient for planarity (correct answer)
Explanation: This question tests your understanding of logical fallacies, specifically the difference between necessary and sufficient conditions in mathematical reasoning. The argument commits the fallacy of affirming the consequent. Here's why: Euler's formula (V−E+F=2V - E + F = 2) is a necessary condition for planarity—every planar graph must satisfy it. However, it's not a sufficient condition—satisfying the formula doesn't guarantee planarity. The argument incorrectly assumes that because graph GG satisfies Euler's formula, it must be planar. This is like saying "All cats are mammals, this animal is a mammal, therefore it's a cat." Option A is incorrect because "denying the antecedent" refers to the fallacy of concluding that if the premise is false, the conclusion is false. That's not happening here—we're reasoning from a true consequent to assume a true antecedent. Option B is wrong because Euler's formula is not equivalent to planarity. Many non-planar graphs can satisfy V−E+F=2V - E + F = 2 when you artificially assign face counts. Option C misidentifies the issue. While the face count does assume planarity in practice, the logical flaw isn't circular reasoning—it's the incorrect assumption that satisfying Euler's formula proves planarity. Option D correctly identifies this as affirming the consequent, since the argument treats a necessary condition as if it were sufficient. Study tip: When evaluating mathematical proofs, always ask whether you're reasoning from sufficient conditions (valid) or just necessary ones (potentially invalid). Necessary conditions tell you what must be true; sufficient conditions tell you what guarantees truth.

Question 16

Analyze this argument: "All recursive algorithms terminate if they have a well-founded base case. The QuickSort algorithm has a base case for arrays of length 1. The algorithm recursively calls itself on smaller subarrays. Therefore, QuickSort always terminates." What logical gap affects this argument's validity?

  1. The argument is valid since recursive calls on smaller inputs with a base case guarantee termination
  2. Missing premise: we need to establish that the recursive calls actually decrease the problem size in every execution path (correct answer)
  3. Affirming the consequent, since having a base case doesn't necessarily imply the algorithm is well-founded
  4. Circular reasoning, since we assume the algorithm terminates in order to prove it terminates
Explanation: The argument has a missing premise. While QuickSort has a base case and makes recursive calls, the validity depends on whether the recursive calls always decrease the problem size. In worst-case scenarios (e.g., already sorted array with poor pivot selection), QuickSort might make unbalanced partitions where one subarray is only one element smaller, leading to O(n2)O(n^2) behavior, though it still terminates. The logical gap is not proving that partitioning always produces smaller subproblems. Choice A assumes what needs to be proven. Choice C misidentifies the logical structure. Choice D incorrectly identifies circular reasoning.

Question 17

Consider the argument: "If a relation is both reflexive and transitive, then it is a preorder. Relation RR on set SS is reflexive since (a,a)∈R(a,a) \in R for all a∈Sa \in S. Also, RR is transitive because whenever (a,b)∈R(a,b) \in R and (b,c)∈R(b,c) \in R, we have (a,c)∈R(a,c) \in R. Therefore, RR is a preorder." What logical rule validates this conclusion?

  1. Modus ponens, since we've established both conditions required for the antecedent
  2. Universal instantiation followed by existential generalization on the relation properties
  3. Hypothetical syllogism, since we're chaining the reflexive and transitive properties together
  4. Conjunction introduction followed by modus ponens, since we combine the two properties then apply the conditional (correct answer)
Explanation: When analyzing logical arguments in discrete math, you need to identify the specific inference rules being applied step by step. This question tests your ability to recognize how formal logic rules work together to reach a conclusion. The argument follows a clear two-step process. First, we have two separate facts: RR is reflexive and RR is transitive. To use these in our conditional statement "If a relation is both reflexive and transitive, then it is a preorder," we must combine them into a single compound statement. This requires conjunction introduction, which allows us to form "RR is reflexive AND RR is transitive" from the two individual statements. Next, we apply modus ponens: since we have the conditional "If (reflexive AND transitive), then preorder" and we've established the antecedent "(reflexive AND transitive)" through conjunction introduction, we can validly conclude that RR is a preorder. Answer D correctly identifies this two-step logical process. Answer A is incomplete because it mentions modus ponens but ignores the crucial conjunction introduction step needed first. Answer B incorrectly focuses on universal and existential quantifiers, which aren't relevant to this propositional logic argument. Answer C misidentifies hypothetical syllogism, which involves chaining conditionals (if PP then QQ, if QQ then RR, therefore if PP then RR), not combining premises to satisfy a conditional's antecedent. Study tip: When identifying logical rules, trace through each step methodically. Look for how premises are combined or transformed before being applied to conditionals—compound antecedents often require conjunction introduction first.

Question 18

Examine this argument: "Every binary tree with nn internal nodes has exactly n+1n+1 leaves. Tree TT has 7 internal nodes. Every node in TT is either internal or a leaf. Therefore, TT has 8 leaves and 15 total nodes." Which aspect of this reasoning requires the most careful justification?

  1. The application of modus ponens is valid, but we must verify that TT is actually a binary tree
  2. The conclusion about total nodes assumes the partition of nodes is exhaustive and mutually exclusive (correct answer)
  3. The reasoning is invalid because we're affirming the consequent about the number of leaves
  4. The argument assumes without justification that every tree with this node structure is binary
Explanation: The most critical justification needed is that the partition "every node is either internal or a leaf" is both exhaustive (covers all nodes) and mutually exclusive (no node is both). While this seems obvious, it's the logical foundation for concluding that total nodes = internal nodes + leaves = 7 + 8 = 15. The modus ponens application (if binary tree with nn internal nodes, then n+1n+1 leaves) is straightforward once we accept TT is binary. Choice A raises a valid point but isn't the most critical logical step. Choice C incorrectly identifies the logical structure. Choice D is less fundamental than establishing the node partition.

Question 19

Consider the argument: "All students who study discrete math also study calculus. Some students who study calculus also study physics. Therefore, some students who study discrete math also study physics." What is the logical status of this argument?

  1. Valid, because the premises guarantee the conclusion through universal and existential instantiation
  2. Invalid, because the middle term 'calculus students' is not distributed in both premises
  3. Invalid, because we cannot conclude anything definite about the intersection of discrete math and physics students (correct answer)
  4. Valid, because if all discrete math students study calculus, and some calculus students study physics, transitivity applies
Explanation: This argument is invalid. Let DD = discrete math students, CC = calculus students, PP = physics students. We have D⊆CD \subseteq C and C∩P≠∅C \cap P \neq \emptyset. However, this doesn't guarantee D∩P≠∅D \cap P \neq \emptyset. The physics students could be entirely from the calculus students who don't study discrete math. Choice A incorrectly applies logical rules. Choice B uses syllogistic terminology incorrectly. Choice D incorrectly applies transitivity to quantified statements.

Question 20

Consider the argument: "Every integer that is divisible by 6 is divisible by both 2 and 3. The number 18 is divisible by both 2 and 3. Since 18=6×318 = 6 \times 3, we can conclude that 18 is divisible by 6." What makes this argument valid or invalid?

  1. Valid by modus ponens, since we've established the antecedent condition for 18
  2. Invalid due to affirming the consequent, despite the additional calculation showing divisibility by 6
  3. Valid because the calculation 18=6×318 = 6 \times 3 provides independent verification of the conclusion (correct answer)
  4. Invalid because being divisible by 2 and 3 is necessary but not sufficient for divisibility by 6
Explanation: The argument is valid, but not due to the logical structure of the first part alone. The statement "18 is divisible by both 2 and 3" would normally be affirming the consequent. However, the calculation 18=6×318 = 6 \times 3 provides direct, independent proof that 18 is divisible by 6. This additional evidence makes the overall argument valid. Choice A misidentifies the logical structure. Choice B ignores the calculation. Choice D is wrong because divisibility by both 2 and 3 actually is sufficient for divisibility by 6 when the numbers are coprime.