Recap B — Set Operations

Question: $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$. Find $A\cup B$ and $A\cap B$.

Answer:

• $A\cup B$: everything from both, each once → $\{1,2,3,4,5\}$
• $A\cap B$: only what is in both → $\{3,4\}$

Question: Using the same sets, find $A\setminus B$ and $B\setminus A$.

Answer:

• $A\setminus B$: start with A, remove anything in B → $\{1,2\}$
• $B\setminus A$: start with B, remove anything in A → $\{5\}$

The order matters: $A\setminus B\ne B\setminus A$.

Question: $U=\{1,2,3,4,5,6,7,8\}$ and $A=\{2,4,6,8\}$. Find $A^{\prime }$ and $n(A^{\prime })$.

Answer:

$A^{\prime }$ is everything in U that is not in A: $\{1,3,5,7\}$, so $n(A^{\prime })=4$.

Question: Is $\{\}=\{\varnothing \}$?

Answer:

No. $\{\}$ has $0$ elements. $\{\varnothing \}$ has $1$ element (the empty set). Sets with different numbers of elements cannot be equal.

Question: $A=\{2,4,6,8\}$. Which are true? (a) $\{2,4\}\subseteq A$ (b) $\varnothing \subseteq A$ (c) $\{4,9\}\subseteq A$

Answer:

• (a) True — $2$ and $4$ are both in A.
• (b) True — the empty set is a subset of every set.
• (c) False — $9$ is not in A, so not every element is in A.

Question: $G=\{1,3,5\}$ and $H=\{\{1\},\{3\},\{5\}\}$. Is $\{1\}\subseteq G$? Is $\{1\}\subseteq H$? Find $G\cap H$.

Answer:

• G is a box of numbers. H is a box of small boxes.
• $\{1\}\subseteq G$: true — the number $1$ is in G.
• $\{1\}\subseteq H$: false — the number $1$ is not in H. H contains the box $\{1\}$, so instead $\{1\}\in H$ is true.
• $G\cap H=\varnothing$ — numbers and boxes are never the same thing.

Question: Using G and H above, find $H\cup \{7\}$, $G\cup \{\varnothing \}$, $H\setminus G$ and $n(G\cup \{\varnothing \})$.

Answer:

• $H\cup \{7\}=\{\{1\},\{3\},\{5\},7\}$
• $G\cup \{\varnothing \}=\{1,3,5,\varnothing \}$
• $H\setminus G=H$ (nothing in H is in G)
• $n(G\cup \{\varnothing \})=4$ — the empty set counts as one element.

Question: $n(A)=20$, $n(B)=15$, $n(A\cup B)=28$. Find $n(A\cap B)$.

Answer:

$28=20+15-n(A\cap B)$

$n(A\cap B)=35-28=7$

Sense check: $7$ is not more than the smaller set ($15$) ✓

Question: $n(A)=25$, $n(B)=18$, $n(A\cap B)=6$. Find $n(A\cup B)$.

Answer:

$n(A\cup B)=25+18-6=37$

Question: $n(A\cup B)=40$, $n(A)=22$, $n(A\cap B)=9$. Find $n(B)$.

Answer:

$40=22+n(B)-9=13+n(B)$, so $n(B)=27$.

Question: $n(A)=10$, $n(B)=24$, $n(A\cup B)=15$. Find $n(A\cap B)$.

Answer:

Formula: $10+24-15=19$. But $A\cap B$ is part of A, so it can have at most $10$ elements — and $n(A\cup B)$ cannot be smaller than $n(B)=24$. The data is impossible. Always check that your answer makes sense, and say so if it does not.

Question: $50$ people were asked about drinks: $30$ drink tea, $25$ drink coffee and $12$ drink both. How many drink tea only, coffee only, and neither?

Answer:

Fill the Venn diagram from the middle outwards:

• 1Both: $12$
• 2Tea only: $30-12=18$
• 3Coffee only: $25-12=13$
• 4Neither: $50-(18+12+13)=50-43=7$

Question: $100$ students were asked about three subjects: Maths (M) $40$, Science (S) $35$, English (E) $30$; M and S $12$; M and E $10$; S and E $8$; all three $5$. Fill in the Venn diagram.

Answer:

Check: $n(M\cup S\cup E)=40+35+30-12-10-8+5=80$ ✓

Question: Using the completed diagram, how many students take (a) Maths only, (b) Maths and Science but not English, (c) none, (d) exactly one subject, (e) at least two subjects?

Answer:

• (a) Maths only: $23$
• (b) M and S only: $7$
• (c) None: $20$
• (d) Exactly one: $23+20+17$ = $60$
• (e) At least two: $7+5+3+5$ = $20$

$1$. $A\cup B=\{2,4,6,8,10\}$; $A\cap B=\{6,8\}$; $A\setminus B=\{2,4\}$

$2$. True — the empty set is a subset of every set.

$3$. $12+9-4=17$

$4$. S and E only = $8-5=3$