Operations on Sets and Venn Diagrams

Putting sets together#

Just as num­bers can be added and mul­ti­plied, sets can be com­bined. Union gath­ers every­thing, inter­sec­tion keeps only what is shared, dif­fer­ence takes some­thing away, and the com­ple­ment is what­ever lies out­side. We will define each of these, pic­ture them in Venn dia­grams, prove the main laws (De Mor­gan's among them), and then use the very prac­ti­cal count­ing for­mula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

Oper­a­tions#

• Union $A \cup B = \{x : x \in A \text{ or } x \in B\}$.
• Inter­sec­tion $A \cap B = \{x : x \in A \text{ and } x \in B\}$. If $A \cap B = \varnothing$, $A$ and $B$ are dis­joint.
• Dif­fer­ence $A - B = \{x : x \in A, x \notin B\}$.
• Com­ple­ment $A' = U - A$.

Laws#

• Com­mu­ta­tive: $A \cup B = B \cup A$, $A \cap B = B \cap A$.
• Asso­cia­tive: $(A \cup B) \cup C = A \cup (B \cup C)$, like­wise for $\cap$.
• Iden­tity: $A \cup \varnothing = A$, $A \cap U = A$.
• Idem­po­tent: $A \cup A = A$, $A \cap A = A$.
• Dis­trib­u­tive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
• Com­ple­ment: $A \cup A' = U$, $A \cap A' = \varnothing$, $(A')' = A$, $\varnothing' = U$.
• De Mor­gan: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

Most of these are obvi­ous once you shade them. De Mor­gan's laws are the ones stu­dents tend to mud­dle, so it is worth see­ing why they hold.

Proof of the first De Mor­gan law. Fol­low a sin­gle ele­ment through the chain: $x \in (A \cup B)' \iff x \notin A \cup B \iff x \notin A \text{ and } x \notin B \iff x \in A' \cap B'$.

Count­ing#

For finite sets, if you sim­ply add the two counts, the ele­ments lying in both get counted twice. Sub­tract them once and the count comes out right:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

With three sets the same idea runs one step fur­ther. Take away the pair­wise over­laps, then add back the cen­tre, which those sub­trac­tions removed once too often:

$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C).$$

Prac­tice ques­tions#

1. $A = \{2, 4, 6, 8, 10\}$, $B = \{5, 10, 15\}$. Find $A \cup B$, $A \cap B$, $A - B$, $B - A$.
2. $U = \{x \in \mathbf{N} : x \le 10\}$, $A = \{x : x \text{ is prime}\}$, $B = \{x : x \text{ is odd}\}$. Find $A'$, $B'$, $(A \cup B)'$, $A' \cap B'$.
3. Are $\{x : x \text{ is a multiple of } 4\}$ and $\{x : x \text{ is odd}\}$ dis­joint?
4. Shade in Venn dia­grams: $A' \cap B$; $(A \cap B)'$; $A - (B \cup C)$.
5. Ver­ify $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ for $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5\}$, $C = \{1, 5, 6\}$.
6. Prove: $A - B = A \cap B'$.
7. Show that $A \cup B = A \cap B$ implies $A = B$.
8. $n(A) = 25$, $n(B) = 18$, $n(A \cup B) = 36$. Find $n(A \cap B)$, $n(A - B)$.
9. Of $60$ stu­dents, $38$ like tea, $30$ like cof­fee, $6$ like nei­ther. How many like both?
10. In a sur­vey of $400$ fam­i­lies: $210$ own a scooter, $180$ a car, $120$ a cycle; $80$ own scooter and car, $50$ car and cycle, $60$ scooter and cycle, $30$ all three. How many own none?

Answers#