Combined Operations on Sets

Question 1

If $\xi=\{1,2,3,4,5,6\}$ and $A=\{2,4,6\}$, then $A' =$

Accepted Answer

$\{1,3,5\}$ — $A'$ contains the elements of $\xi$ not in $A$: $\{1,3,5\}$.

Question 2

The complement $A'$ consists of the elements of the universal set that are:

Accepted Answer

Not in $A$ — By definition, $A'$ is everything in $\xi$ that is not in $A$.

Question 3

Using $n(A\cup B)=n(A)+n(B)-n(A\cap B)$: if $n(A)=12$, $n(B)=9$ and $n(A\cup B)=15$, find $n(A\cap B)$.

Accepted Answer

6 — $n(A\cap B)=n(A)+n(B)-n(A\cup B)=12+9-15=6$.

Question 4

If $\xi=\{1,2,\ldots,10\}$, $A=\{1,2,3,4,5\}$ and $B=\{4,5,6,7\}$, find $(A\cap B)'$.

Accepted Answer

$\{1,2,3,6,7,8,9,10\}$ — $A\cap B=\{4,5\}$, so its complement is everything else: $\{1,2,3,6,7,8,9,10\}$.

Question 5

If $n(\xi)=40$ and $n(A)=25$, then $n(A') =$

Accepted Answer

15 — $n(A')=n(\xi)-n(A)=40-25=15$.

Question 6

By De Morgan's law, $A' \cap B' =$

Accepted Answer

$(A\cup B)'$ — De Morgan's law: $A'\cap B' = (A\cup B)'$.

Question 7

If $n(A)=20$, $n(B)=15$ and $A$ and $B$ are disjoint, find $n(A\cup B)$.

Accepted Answer

35 — Disjoint means $n(A\cap B)=0$, so $n(A\cup B)=20+15-0=35$.

Question 8

In a class of $35$ students, $20$ study Physics, $18$ study Chemistry and $8$ study both. How many study neither?

Accepted Answer

5 — $n(P\cup C)=20+18-8=30$, so neither $=35-30=5$.

Question 9

If $\xi=\{	ext{integers from }1	ext{ to }9\}$ and $A=\{	ext{odd numbers}\}$, then $A' =$

Accepted Answer

$\{2,4,6,8\}$ — $A=\{1,3,5,7,9\}$, so $A'$ is the even numbers from $1$ to $9$: $\{2,4,6,8\}$.

Question 10

Given $n(A)=n(B)=n(A\cap B)=10$, find $n(A\cup B)$.

Accepted Answer

10 — $n(A\cup B)=10+10-10=10$. (Here $A$ and $B$ are in fact the same set.)