Operations on Sets — Exercises

Contents

1Basic union, intersection, difference easy

Let $A = \{2, 4, 6, 8\}$ and $B = \{1, 2, 3, 4, 5\}$. Compute: $A \cup B$, $A \cap B$, $A \setminus B$, $B \setminus A$.

2N-ary operations easy

Given the family of sets L = [{1,2,3}, {2,3,4}, {3,4,5}, {3,5,7}], compute:

• $\displaystyle\bigcup_{i} L_i$
• $\displaystyle\bigcap_{i} L_i$

3Complement with a universe easy

With universe $U = \{1, 2, \dots, 10\}$ and $A = \{1, 3, 5, 7, 9\}$ (odd numbers), compute $A'$. Then verify $A \cup A' = U$ and $A \cap A' = \varnothing$.

4Operator precedence medium

Complement > Intersection > Union. With $U = \{1..10\}$, $A = \{1,2,3\}$, $B = \{2,3,4\}$, $C = \{3,4,5\}$, evaluate by hand, then in Python:

1. $A \cup B \cap C$  (meaning $A \cup (B \cap C)$)
2. $(A \cup B) \cap C$
3. $A' \cap B \cup C$  (meaning $(A' \cap B) \cup C$)

5The 8 regions of a 3-set Venn medium

With $A = \{1..7\}$, $B = \{3..9\}$, $C = \{5..11\}$, write a function that returns the contents of each of the 8 regions of the Venn diagram of $A, B, C$. Which regions are empty?

6Inclusion–Exclusion (2 sets) medium

Pick any $A, B \subseteq \{1, \dots, 20\}$ randomly and verify $|A \cup B| = |A| + |B| - |A \cap B|$ holds on 1000 random pairs.

7De Morgan by brute force medium

For $U = \{1, 2, 3, 4\}$, check $(A \cup B)' = A' \cap B'$ for every pair of subsets of $U$ (there are $2^4 \times 2^4 = 256$ pairs).

8Distributivity check medium

Prove (by exhaustive check on $U = \{1, 2, 3\}$) that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ for all $A, B, C \subseteq U$.

9Membership table medium

Print the full membership (truth) table for $(A \cup B)'$ and $A' \cap B'$ and show they are equal.

10Simplify with SymPy hard

Using SymPy's simplify (boolean), show that $A \cup (A \cap B) \equiv A$ (absorption).

Then show $(A \cap B) \cup (A \cap B') \equiv A$ (a useful "splitting" identity).

11Symmetric difference identities hard

Show, by brute force on $U = \{1, 2, 3, 4\}$, that:

1. $A \,\triangle\, B = (A \cup B) \setminus (A \cap B)$
2. $A \,\triangle\, A = \varnothing$
3. $A \,\triangle\, B \,\triangle\, C$ is associative

12Word problem — 3-set Venn hard

A survey of 200 people about three streaming services (N, P, D) reports:

• 120 use N, 90 use P, 70 use D.
• 50 use N and P, 30 use N and D, 25 use P and D.
• 15 use all three.

How many use:

1. at least one service?
2. exactly one service?
3. exactly two services?
4. none?

Going further

• Generalize venn_regions from 3 to $n$ sets.
• Visualize exercise 12 with matplotlib_venn.venn3.
• Re-prove distributivity using sympy.simplify(Equivalent(...)) instead of brute force.
• Prove $A \,\triangle\, B = (A \cap B') \cup (A' \cap B)$ symbolically with SymPy.

Open the main lesson for the matching theory.