Sets, Venn Diagrams, and Overlap - Quantitative Reasoning

1. What is the relationship between $A\setminus B$ and $A\cap B^c$?

   $A\setminus B=A\cap B^c$. Expresses set difference as the intersection of A with the complement of B.

2. What is the definition of the symmetric difference $A\triangle B$?

   $A\triangle B=(A\setminus B)\cup(B\setminus A)$. Defines the symmetric difference as the union of elements in A but not B and elements in B but not A.

3. Find $|A\cup B|$ if $|U|=50$ and $|(A\cup B)^c|=12$.

   $38$. Determines union size by subtracting the complement from the universal set: $50 - 12 = 38$.

4. State the formula for the size of a union: $|A\cup B|$ in terms of $|A|,|B|,|A\cap B|$.

   $|A\cup B|=|A|+|B|-|A\cap B|$. Applies the inclusion-exclusion principle to count elements in the union without double-counting the intersection.

5. Find $|A\triangle B|$ if $|A|=14$, $|B|=11$, and $|A\cap B|=5$.

   $15$. Computes symmetric difference as sum of exclusive parts: $(14-5) + (11-5) = 15$.

6. Find $|B\setminus A|$ if $|B|=19$ and $|A\cap B|=6$.

   $13$. Computes set difference as elements only in B: $19 - 6 = 13$.

7. Find $|A^c|$ if $|U|=60$ and $|A|=22$.

   $38$. Calculates the complement as elements not in A: $60 - 22 = 38$.

8. What is the definition of the intersection $A \cap B$ in set notation?

   $A \cap B=\{x: x\in A\text{ and }x\in B\}$. Defines the set containing all elements that belong to both A and B.

9. Find $|A\cap B\cap C|$ if $|A\cap B|=9$, $|A\cap B\setminus C|=6$.

   $3$. Determines the triple intersection by subtracting the part excluding C: $9-6=3$.

10. What does it mean to say $A\subseteq B$?

    Every element of $A$ is in $B$. Indicates that A is a subset of B, meaning all elements of A are also elements of B.

11. Find $|A\cap B|$ if $|A|=20$, $|B|=17$, and $|A\cup B|=30$.

    $7$. Solves for intersection using inclusion-exclusion rearranged: $20 + 17 - 30 = 7$.

12. Find $|A\setminus B|$ if $|A|=25$ and $|A\cap B|=9$.

    $16$. Computes set difference as elements only in A: $25 - 9 = 16$.

13. Find $|U\setminus(A\cup B)|$ if $|U|=80$, $|A|=35$, $|B|=30$, and $|A\cap B|=10$.

    $25$. Calculates elements outside the union: $80 - (35+30-10)=25$.

14. Find $|A\cup B|$ if $|A|=18$, $|B|=15$, and $|A\cap B|=7$.

    $26$. Calculates the union size using inclusion-exclusion: $18 + 15 - 7 = 26$.

15. What is the definition of the complement $A^c$ relative to universal set $U$?

    $A^c=U\setminus A=\{x\in U: x\notin A\}$. Defines the set of elements in the universal set U that are not in A.

16. What is the definition of the set difference $A\setminus B$?

    $A\setminus B=\{x: x\in A\text{ and }x\notin B\}$. Defines the set of elements in A but not in B.

17. What is the definition of the union $A \cup B$ in set notation?

    $A \cup B=\{x: x\in A\text{ or }x\in B\}$. Defines the set containing all elements that belong to A or B or both.

18. State De Morgan's law for the complement of an intersection: $(A\cap B)^c$.

    $(A\cap B)^c=A^c\cup B^c$. States De Morgan's law, equating the complement of an intersection to the union of complements.

19. What does it mean to say two sets $A$ and $B$ are disjoint?

    $A\cap B=\varnothing$. Indicates that A and B have no elements in common, so their intersection is the empty set.

20. State the 3-set inclusion–exclusion formula for $|A\cup B\cup C|$.

    $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$. Applies the inclusion-exclusion principle for three sets to count the total unique elements in the union.

21. State De Morgan's law for the complement of a union: $(A\cup B)^c$.

    $(A\cup B)^c=A^c\cap B^c$. States De Morgan's law, equating the complement of a union to the intersection of complements.

22. Identify the set represented by "in exactly one of $A$ or $B$" using set notation.

    $(A\setminus B)\cup(B\setminus A)$. Represents elements that are in exactly one of the sets A or B, forming the symmetric difference.

23. Find $|A\cap B^c|$ if $|A|=16$ and $|A\cap B|=9$.

    $7$. Calculates elements in A but not B: $16 - 9 = 7$.

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