Logic

Sets and Set Operations

Lesson

A set is an unordered collection of distinct elements. Three operations build new sets from old ones.

Three operations

Union $A \cup B$ : everything in A or B (or both).
Intersection $A \cap B$ : elements in both A and B.
Complement $A'$ : elements in the universe NOT in A.
Difference $A \setminus B$ : in A but not in B.

Inclusion-exclusion

|A \cup B| = |A| + |B| - |A \cap B|

Power set

A set of size $n$ has $2^n$ subsets.

Problems use universe $U = \{1, 2, \dots, 10\}$ , set $A = \{1, 2, 3, 4, 5\}$ , and set $B = \{4, 5, 6, 7, 8\}$ unless stated otherwise.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

|A \cap B|

Problem 2

|A \cup B|

Problem 3

|A|

Problem 4

|B|

Practice

Standard problems matching the lesson.

Problem 5

|A'| \text{ (complement in } U\text{)}

Problem 6

|B'|

Problem 7

|A \setminus B|

Problem 8

|B \setminus A|

Problem 9

|A \cup B'|

Problem 10

|A \cap B'|

Problem 11

|A \cup B \cup \{9\}|

Problem 12

|U|

Problem 13

\text{Number of subsets of } A

Problem 14

\text{Number of size-2 subsets of } A

Problem 15

|\emptyset|

Problem 16

|\{x \in U : x \text{ even}\}|

Problem 17

|\{x \in U : x \text{ multiple of 3}\}|

Problem 18

|A \cap \{x \in U : x \text{ even}\}|

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

C = \{1, 2\}. \ |A \cup B \cup C|

Problem 20

|A \cap B \cap \{5, 6, 7\}|

Problem 21

|A| + |B| - |A \cap B|

Problem 22

|X|=10, |Y|=8, |X \cap Y|=3. \ |X \cup Y|?

Problem 23

X \cap Y = \emptyset, |X|=4, |Y|=6. \ |X \cup Y|?

Problem 24

\text{Power set of size-3 set has how many elements?}

Problem 25

|X|=20, |Y|=15, |X \cup Y|=30. \ |X \cap Y|?

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