Set Theory Basics - Universal Set, Union, Intersection, and Sieve Principle

50 practice flashcards covering universal set, basic operations (union, intersection, difference, complement), disjointness, and inclusion-exclusion with numerical examples.

What is the universal set?

The universal set, denoted U, is the set that contains all objects under study; every set considered is a subset of U.

In basic set operations, what is the relationship of all sets to the universal set?

All sets under investigation are subsets of the universal set U.

What is the union of two sets A and B?

The union A ∪ B is the set of elements that belong to A or to B or to both.

What is the intersection of two sets A and B?

The intersection A ∩ B is the set of elements that are common to both A and B.

What is the difference A − B?

The set of elements that are in A but not in B.

What is the complement A′ (relative to U)?

The set of elements in the universal set U that do not belong to A.

What is the Sieve Principle also known as?

The Inclusion–Exclusion Principle for counting elements in unions.

How many multiples of 2 are there from 1 to 50?

25.

How many multiples of 3 are there from 1 to 50?

16.

How many multiples of 4 are there from 1 to 50?

12.

How many numbers up to 50 are multiples of both 2 and 3?

8 (multiples of 6).

How many numbers up to 50 are multiples of both 2 and 4?

12 (multiples of 4).

How many numbers up to 50 are multiples of both 3 and 4?

4 (multiples of 12).

How many numbers up to 50 are multiples of 2, 3, or 4?

33.

Are the empty set ∅ and any set disjoint?

Yes; ∅ has no elements, so its intersection with any set is ∅.

Are consonants and vowels in the English alphabet disjoint?

Yes; a letter cannot be both a consonant and a vowel.

Are black-suit cards and aces disjoint in a standard deck?

No; there are aces that are of black suits (e.g., ace of spades and ace of clubs).

What is an example of a universal set for fractions?

If the universal set is all numbers, then the set of fractions is a subset of that universal set.

What does A ∪ B represent in a Venn diagram?

The region containing all elements that are in A or in B or in both.

What does A ∩ B represent in a Venn diagram?

The overlapping region where A and B intersect.

What is the complement of the empty set ∅ within U?

The complement of ∅ is U.

If U = {0,1,2,3,4,5,6,7,8,9}, what is the complement of {0,1,2,3,4,5,6,7,8,9}?

∅ (the complement is empty because the set is the entire universe).

Using U = {0,1,2,3,4,5,6,7,8,9}, what is the complement of {1,3,5,7,9}?

{0,2,4,6,8}.

What is the two-set inclusion-exclusion formula for |A∪B|?

|A∪B| = |A| + |B| − |A∩B|.

What is the three-set inclusion-exclusion formula for |A∪B∪C|?

|A∪B∪C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.

Given A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}, what is A∪B?

{1,2,3,4,5,6,7,8}.

Given A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}, what is A∩B?

∅.

Given A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}, what is A∩C?

{6}.

Given A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}, what is B∩C?

{3}.

Compute (A∪B) ∩ C using the sets above.

{3,6}.

Compute (A∩B) ∪ C using the sets above.

{3,6,9}.

Compute A ∪ (B ∪ C) with the above sets.

{1,2,3,4,5,6,7,8,9}.

Compute A ∩ (B ∪ C) with the above sets.

{6}.

What is A − B for A = {2,4,6,8} and B = {1,3,5,7}?

{2,4,6,8}.

What is (A ∪ B) − C for A,B,C as above?

{1,2,4,5,7,8}.

What is A − (B ∩ C) for A,B,C as above?

{2,4,6,8}.

What is the complement of A in U when U = {0,…,9} and A = {0,…,9}?

∅.

What is the complement of B = {1,3,5,7,9} in U = {0,…,9}?

{0,2,4,6,8}.

What is (A ∪ B)′ when A = U and B = {1,3,5,7,9}?

∅.

What is (A ∩ B)′ when A = U and B = {1,3,5,7,9}?

{0,2,4,6,8}.

What does the symbol ∪ denote?

Union of sets.

What does the symbol ∩ denote?

Intersection of sets.

What does the symbol ′ denote?

Complement.

What does the symbol ∅ denote?

The empty set.

What does the symbol U denote?

The universal set.

State the two-set inclusion-exclusion formula for |A∪B| (repeat for clarity).

|A∪B| = |A| + |B| − |A∩B|.

State the three-set inclusion-exclusion formula for |A∪B∪C| (repeat for clarity).

|A∪B∪C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.

If |A| = 25, |B| = 16, and |A∩B| = 8, what is |A∪B|?

33.

If |A| = 25, |B| = 16, |C| = 12, |A∩B| = 8, |A∩C| = 12, |B∩C| = 4, and |A∩B∩C| = 4, what is |A∪B∪C|?

33.

What is |A∩B| for the sets A = {2,4,6,8} and B = {1,3,5,7}?

0 (empty set).

What is |A∩C| for the sets A = {2,4,6,8} and C = {3,6,9}?

{6}.

What is |B∩C| for the sets B = {1,3,5,7} and C = {3,6,9}?

{3}.

What is the value of (A∪B) ∩ C with A,B,C as above?

{3,6}.

What is the value of (A∩B) ∪ C with A,B,C as above?

{3,6,9}.

What is the result of A ∪ (B ∪ C) for A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}?

{1,2,3,4,5,6,7,8,9}.

What is the result of A ∩ (B ∪ C) for A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}?

{6}.

What is A − B when A = {2,4,6,8} and B = {1,3,5,7} (again as a quick check)?

{2,4,6,8}.

What is (A ∪ B) − C for A,B,C as above (repeat for practice)?

{1,2,4,5,7,8}.

What is A − (B ∩ C) for A,B,C as above (repeat for practice)?

{2,4,6,8}.

What is the complement of A when A = {0,1,2,3,4,5,6,7,8,9} and U = {0,…,9}?

∅.

What is the complement of B = {1,3,5,7,9} in U = {0,…,9}?

{0,2,4,6,8}.

What is the complement of A ∪ B in the previous setup if A = U and B = {1,3,5,7,9}?

∅.

What is the complement of A ∩ B when A = U and B = {1,3,5,7,9}?

{0,2,4,6,8}.

What is the Sieve Principle used for in counting?

Counting the number of elements in unions of finite sets using inclusion-exclusion.

If A is the set of multiples of 2 up to 50 and B of 3 up to 50, what is |A|?

25.

If A is the set of multiples of 2 up to 50 and B of 3 up to 50, what is |B|?

16.

If A is the set of multiples of 2 up to 50 and B of 3 up to 50, what is |A∩B| (multiples of 6 up to 50)?

8.

Using inclusion-exclusion, what is |A∪B| for the multiples example?

33.

Using A = {2,4,6,8}, B = {1,3,5,7}, C = {3,6,9}, what is |A∪B∪C|?

33.

Which notation indicates the universal set?

U.

Which notation indicates the empty set?

∅.

Which notation indicates the complement of a set?

′ (prime symbol) or the word complement.

What does A ⊆ B signify?

A is a subset of B.

What is the result of A ∪ ∅?

A.

What is the result of A ∩ U?

A.

In a Venn diagram with A and B, what region does A ∪ B include?

All regions belonging to A, B, or both.

In a Venn diagram with A and B, what region does A ∩ B include?

The overlapping region of A and B.

What is the difference between A and B if B contains all elements of A?

A − B = ∅.

Are the sets {1,2} and {2,3} disjoint?

No; they share element 2 (A ∩ B = {2}).