Finite Mathematics : Logic, Sets, and Counting

Example Questions

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Example Question #1 : Logic, Sets, And Counting

If

what is

?

Explanation:

To solve this problem first identify what the notation means and what exactly the question is asking.

means the number of elements in the set  is seven.

means the number of elements in the set  is twelve.

means the number of elements that exist in both  and  is the empty set, thus none of the elements that are in  are in .

Now the question asks to find,

which means to find the number of unique elements that exist in  and in .

Since the intersection of the two sets is the empty set, that means all elements in  and in  are unique.

Thus calculating the answer is as follows,

Example Question #2 : Logic, Sets, And Counting

Let  represent the situation that "The sun is shinning" and  represents the situation "I got sunburnt".

In words, what does  mean?

The sun is shinning and I got sunburnt.

The sun is shinning and I didn't get sunburnt

None of the options.

The sun is not shinning and I got sunburnt.

It is not the case that the sun is shinning and I got sunburnt.

It is not the case that the sun is shinning and I got sunburnt.

Explanation:

First, identify and understand the notation.

means the intersection between the two sets. The intersection is when both situations overlap and are true.

means "not the case"

Therefore

means "It is not the case that the sun is shinning and I got sunburnt."

Example Question #3 : Logic, Sets, And Counting

If

what is

?

Explanation:

To solve this problem first identify what the notation means and what exactly the question is asking.

means the number of elements in the set  is seven.

means the number of elements in the set  is twelve.

means the number of elements that exist in both  and  are three.

Now the question asks to find,

which means to find the number of unique elements that exist in  and in .

Since the intersection of the two sets is three,

Calculating the answer is as follows,

Example Question #1 : Logic, Sets, And Counting

Given: Sets and such that

Which of the following is a true statement?

and are disjoint sets.

and cannot exist.

and are not disjoint sets.

and are not disjoint sets.

Explanation:

and are defined to be disjoint sets if their intersection has no elements. This occurs if and only if

;

that is, if and only if the number of elements in the union is equal to the sum of the elements in the sets.

By substitution, we see that this is the statement

,

so and exist, and are not disjoint.

Example Question #1 : Logic, Sets, And Counting

Let be the set of the ten best Presidents of the United States.

True or false: is an example of a well-defined set.

False

True

False

Explanation:

A set is well-defined if is defined in a way that makes it clear what elements are and are not in the set. The "ten best Presidents of the United States" is open to the judgment of whoever defines the set, so it is ambiguous which Presidents belong to that set.

Example Question #1 : Logic, Sets, And Counting

True or false: The sentence "7 is a composite number" is a logical statement.

False

True

True

Explanation:

A logical statement is a sentence that can be determined to be true or false. 7 is known to not be a composite number, so the sentence is known to be false; that makes it a valid example of a logical statement.

Example Question #6 : Logic, Sets, And Counting

Consider the statement:

"John is a carpenter and Jim is a taxi driver."

True or false: the negation of this is the statement

"John is not a carpenter or Jim is not a taxi driver."

True

False

True

Explanation:

The statement "John is a carpenter and Jim is a taxi driver" is a compound statement, comprising two simple statements connected with an "and". This statement is therefore

and ,

where

John is a carpenter

Jim is a taxi driver.

The negation of a statement is "not "; the negation of " and " is

"Not ( and )"

By DeMorgan's Law, this is equivalent to the statement

"(Not ) or (Not )"

Thus,"John is not a carpenter or Jim is not a taxi driver." is indeed the negation of the given statement.

Example Question #1 : Logic, Sets, And Counting

Examine the above Venn diagram.

Let be the set of all people. Let be the set of people who listen to Band A and be the set of people who listen to Band B. Which of the following describes the shaded portion of the Venn diagram?

The set of all people who do not listen to Band B.

The set of all people who do not listen to Band A.

The set of all people who do not listen to Band A or Band B.

The set of all people who listen to Band A but do not listen to Band B.

The set of all people who listen to Band B but do not listen to Band A.

The set of all people who do not listen to Band A.

Explanation:

The shaded portion of the Venn diagram is exactly the portion of the universal set not in - the complement of . Since is the set of all people who listen to Band A, this complement is the set of all people who do not listen to Band A.

Example Question #1 : Logic, Sets, And Counting

Consider the conditional statement:

"If Andy is a Freemason, then Danny is not a Freemason. "

Which statement is the contrapositive of this statement?

If Andy is a Freemason, then Danny is a Freemason.

If Danny is a Freemason, then Andy is not a Freemason.

If Danny is a Freemason, then Andy is a Freemason.

If Danny is not a Freemason, then Andy is a Freemason.

If Andy is not a Freemason, then Danny is a Freemason.

If Danny is a Freemason, then Andy is not a Freemason.

Explanation:

Let and be the simple statements:

: Andy is a Freemason

: Danny is not a Freemason.

The given conditional is therefore "If then ".

The contrapositive of this conditional is defined to be "If not then not ," which here is the statement "If Danny is a Freemason, then Andy is not a Freemason."

Example Question #9 : Logic, Sets, And Counting

Define to be the set of all smart Australians.

True or false: is an example of a well-defined set.

False

True

False

Explanation:

A set is well-defined if it can be determined with no ambiguity which elements are and are not in the set. In the case of , the word "smart" is ambiguous, since the definition can change according to who is deciding who is "smart" and who is not. is therefore not a well-defined set.

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