Sets and Venn Diagrams

What to remember

• A set is a well-defined collection of distinct objects. "Well-defined" means we can say clearly whether any object belongs to it. "The tall students of a class" is not a set; "the students taller than 150 cm" is a set.
• Four operations build every set problem: union (A ∪ B), intersection (A ∩ B), difference (A − B) and complement (A′). Venn diagrams show them as shaded regions.
• The counting formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) solves most word problems. For a set with n elements, the number of subsets is 2^n.

Basic ideas and notation

A set is written with capital letters and its members (elements) inside curly brackets. We write x ∈ A for "x belongs to A" and x ∉ A for "x does not belong to A". There are three ways to describe a set.

In roster form each element is written once, and the order does not matter. So {1, 2, 3} and {3, 1, 2} are the same set, and {1, 1, 2} is just {1, 2}.

Standard number sets: N (natural), W (whole), Z (integers), Q (rational), R (real). The symbol ϕ or { } stands for the empty set.

Types of sets

• Empty (null) set: has no element, ϕ. Example: the set of prime numbers between 23 and 29. Note that {0} is not empty; it has one element.
• Singleton set: exactly one element, such as {5}.
• Finite set: the elements can be counted and the counting stops. Infinite set: the counting never stops, such as the set of natural numbers.
• Equal sets: A = B when both have exactly the same elements.
• Equivalent sets: same number of elements, n(A) = n(B), though the elements may differ.
• Disjoint sets: no common element, so A ∩ B = ϕ.
• Universal set (U): the set that contains all the objects under discussion. Every set in that problem is a subset of U.
• Subset: A ⊆ B if every element of A is also in B. Proper subset: A ⊂ B if A ⊆ B and A ≠ B.

Facts about subsets: the empty set is a subset of every set; every set is a subset of itself. The power set P(A) is the set of all subsets of A. If n(A) = m, then n(P(A)) = 2^m, and the number of proper subsets is 2^m − 1. Example: A = {1, 2, 3} has 2³ = 8 subsets and 7 proper subsets.

Cardinal number n(A) means the number of elements of a finite set A. The cardinal number of ϕ is 0.

Operations on sets

For A = {1, 2, 3, 4} and B = {3, 4, 5, 6} in U = {1, 2, 3, 4, 5, 6, 7, 8}:

Note that A − B and B − A are usually different. Also A − B = A ∩ B′. The complement of the universal set is ϕ, and the complement of ϕ is U. For any set, (A′)′ = A.

Laws of sets

De Morgan's laws in words: the complement of a union is the intersection of the complements; the complement of an intersection is the union of the complements.

If A ⊆ B, then A ∪ B = B and A ∩ B = A.

Venn diagrams

John Venn introduced the diagrams. The universal set U is drawn as a rectangle, and each set as a circle or closed curve inside it. Two overlapping circles mean the sets have some common elements. Circles drawn apart mean the sets are disjoint. A circle drawn inside another means a subset.

Regions for two sets A and B (four regions in all): only A (A − B), only B (B − A), both (A ∩ B), and neither (outside both circles, which is (A ∪ B)′). Always start by filling the innermost region, the intersection, and then fill outward.

Counting formulas

For finite sets:

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• If A and B are disjoint, n(A ∪ B) = n(A) + n(B).
• n(A − B) = n(A) − n(A ∩ B).
• n(A′) = n(U) − n(A).
• For three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).

Worked example 1: In a class of 40 students, 25 play cricket, 20 play kabaddi and 8 play both. Students playing at least one game = 25 + 20 − 8 = 37. Students playing neither = 40 − 37 = 3. Only cricket = 25 − 8 = 17.

Worked example 2: If n(A) = 12, n(B) = 9 and n(A ∪ B) = 17, then n(A ∩ B) = 12 + 9 − 17 = 4.

Worked example 3: In a group of 60 people, 35 like tea, 30 like coffee and 10 like neither. People who like at least one = 50. So both = 35 + 30 − 50 = 15.

Classroom angle

Start with real collections students can see: the set of students wearing glasses, the set of students with a pencil. Let them stand in two hoops marked on the floor and see the overlap as the intersection. A common error is to think "the set of good players" is a set; stress the idea of a well-defined collection. Another error is to treat ϕ and {ϕ} as the same; {ϕ} has one element, so n({ϕ}) = 1.

Exam traps

• ϕ and {0}: ϕ has no element; {0} has one element.
• ϕ and {ϕ}: n(ϕ) = 0 but n({ϕ}) = 1, and its power set has 2 subsets.
• ⊂ and ∈: ∈ links an element with a set; ⊂ links a set with a set. So 2 ∈ {1, 2} but {2} ⊂ {1, 2}.
• Equal and equivalent sets: equal means the same elements; equivalent only means the same count.
• A − B and B − A: these are not equal in general.
• Subsets and proper subsets: 2^n counts all subsets, 2^n − 1 counts proper subsets.
• De Morgan: the complement changes ∪ to ∩ and ∩ to ∪; it is not simply distributed.
• Repeated elements: {a, a, b} has only two elements.

One-liners

• 1. A set is a well-defined collection of distinct objects.
• 2. Order and repetition do not matter in a set.
• 3. ϕ is a subset of every set.
• 4. A set with n elements has 2^n subsets.
• 5. A set with n elements has 2^n − 1 proper subsets.
• 6. A ∩ B = ϕ means A and B are disjoint.
• 7. n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• 8. A − B = A ∩ B′.
• 9. (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
• 10. A ∪ A′ = U and A ∩ A′ = ϕ.
• 11. If A ⊆ B, then A ∪ B = B.
• 12. Venn diagrams show sets as closed curves inside a rectangle for U.

Practice questions

1. Which of the following is a well-defined set?

   1. The best songs of a decade
   2. The most beautiful flowers
   3. The brave students of a school
   4. The vowels of the English alphabet
   Answer

   D. The vowels of the English alphabet

   A set must be well-defined; only the vowels can be decided clearly.

2. How many subsets does a set with 4 elements have?

   1. 8
   2. 4
   3. 15
   4. 16
   Answer

   D. 16

   2^4 = 16.

3. How many proper subsets does a set with 3 elements have?

   1. 8
   2. 6
   3. 3
   4. 7
   Answer

   D. 7

   2³ − 1 = 7.

4. The set of prime numbers between 23 and 29 is

   1. an infinite set
   2. an empty set
   3. a singleton set
   4. a set with two elements
   Answer

   B. an empty set

   The numbers 24 to 28 are all composite, so the set is empty.

5. The number of elements in the set {ϕ} is

   1. 0
   2. 2
   3. 1
   4. Not defined
   Answer

   C. 1

   {ϕ} contains one element, the empty set.

6. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B is

   1. {1, 2}
   2. {3, 4}
   3. {1, 2, 3, 4, 5, 6}
   4. {5, 6}
   Answer

   B. {3, 4}

   Common elements are 3 and 4.

7. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∪ B has how many elements?

   1. 6
   2. 8
   3. 4
   4. 2
   Answer

   A. 6

   Union = {1, 2, 3, 4, 5, 6}; common elements are counted once.

8. If U = {1, 2, …, 8} and A = {1, 2, 3, 4}, then A′ is

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

   C. {5, 6, 7, 8}

   The complement holds elements of U not in A.

9. If n(A) = 12, n(B) = 9 and n(A ∪ B) = 17, then n(A ∩ B) is

   1. 3
   2. 5
   3. 4
   4. 21
   Answer

   C. 4

   12 + 9 − 17 = 4.

10. In a class of 40, 25 play cricket, 20 play kabaddi and 8 play both. How many play neither game?

    1. 5
    2. 8
    3. 12
    4. 3
    Answer

    D. 3

    At least one = 25 + 20 − 8 = 37; neither = 40 − 37 = 3.

11. In the same class (40 students; 25 cricket, 20 kabaddi, 8 both), how many play only cricket?

    1. 25
    2. 17
    3. 12
    4. 15
    Answer

    B. 17

    Only cricket = 25 − 8 = 17.

12. In a group of 60 people, 35 like tea, 30 like coffee and 10 like neither. How many like both?

    1. 25
    2. 20
    3. 5
    4. 15
    Answer

    D. 15

    At least one = 50; both = 35 + 30 − 50 = 15.

13. If A and B are disjoint, then n(A ∪ B) equals

    1. n(A ∩ B)
    2. n(A) + n(B)
    3. n(A) − n(B)
    4. n(A) × n(B)
    Answer

    B. n(A) + n(B)

    With no common elements, the counts just add.

14. De Morgan's law states that (A ∪ B)′ equals

    1. A′ ∪ B′
    2. A′ − B′
    3. A′ ∩ B′
    4. A ∩ B
    Answer

    C. A′ ∩ B′

    The complement of a union is the intersection of the complements.

15. The set A − B is the same as

    1. A ∩ B′
    2. A′ ∩ B
    3. A ∪ B′
    4. A′ ∩ B′
    Answer

    A. A ∩ B′

    Elements in A and not in B are in A and in B′.

16. Which symbol correctly completes: 2 __ {1, 2, 3}?

    1. =
    2. ∈
    3. ⊂
    4. ⊄
    Answer

    B. ∈

    2 is an element, so ∈ is used.

17. Which statement is correct for A = {1, 2, 3}?

    1. {2} ∈ A
    2. 2 ⊂ A
    3. {2} ⊂ A
    4. ϕ ∈ A
    Answer

    C. {2} ⊂ A

    A set is a subset of a set; 2 alone is an element.

18. The power set of A = {a, b} has

    1. 2 elements
    2. 3 elements
    3. 8 elements
    4. 4 elements
    Answer

    D. 4 elements

    Subsets are ϕ, {a}, {b}, {a, b}, so 2² = 4.

19. If A ⊆ B, then A ∪ B equals

    1. B
    2. U
    3. A
    4. ϕ
    Answer

    A. B

    All elements of A are already in B.

20. Two sets with the same number of elements, but not necessarily the same elements, are called

    1. disjoint sets
    2. universal sets
    3. equal sets
    4. equivalent sets
    Answer

    D. equivalent sets

    Equivalent sets have the same cardinal number.

21. The set builder form of A = {2, 4, 6, 8} is

    1. {x : x is a natural number less than 10}
    2. {x : x is an even natural number less than 10}
    3. {x : x is an even number}
    4. {x : x is a multiple of 4}
    Answer

    B. {x : x is an even natural number less than 10}

    Only the even natural numbers below 10 give 2, 4, 6, 8.

22. If n(U) = 50 and n(A) = 18, then n(A′) is

    1. 32
    2. 18
    3. 68
    4. 30
    Answer

    A. 32

    n(A′) = n(U) − n(A) = 32.

23. If n(A) = 20, n(B) = 15 and n(A ∩ B) = 6, then n(A − B) is

    1. 26
    2. 9
    3. 14
    4. 21
    Answer

    C. 14

    n(A − B) = 20 − 6 = 14.

24. A set has 5 elements. How many of its subsets contain a particular fixed element?

    1. 16
    2. 31
    3. 32
    4. 10
    Answer

    A. 16

    Fix the element; each of the other 4 can be in or out: 2^4 = 16.

25. Set A has 3 elements and set B has 4 elements. The least possible value of n(A ∪ B) is

    1. 3
    2. 7
    3. 4
    4. 0
    Answer

    C. 4

    Least when A ⊆ B, then the union is B with 4 elements.

26. Using the same sets (n(A) = 3, n(B) = 4), the greatest possible value of n(A ∪ B) is

    1. 4
    2. 12
    3. 1
    4. 7
    Answer

    D. 7

    Greatest when they are disjoint: 3 + 4 = 7.

27. In a Venn diagram the universal set is usually drawn as a

    1. straight line
    2. rectangle
    3. single point
    4. triangle
    Answer

    B. rectangle

    U is shown as a rectangle, and the sets as circles inside it.

28. If n(A) = 10, n(B) = 12 and n(A ∩ B) = 5, the number of elements in exactly one of the two sets is

    1. 17
    2. 12
    3. 7
    4. 22
    Answer

    B. 12

    Only A = 5, only B = 7; total 12.

29. Which pair of sets is equal?

    1. {1, 2, 3} and {3, 1, 2, 2}
    2. {1, 2} and {1, 2, 3}
    3. {a} and {{a}}
    4. ϕ and {0}
    Answer

    A. {1, 2, 3} and {3, 1, 2, 2}

    Order and repetition do not matter in a set.

30. Of 100 students, 60 study Maths and 50 study Science, and every student studies at least one. How many study both?

    1. 50
    2. 110
    3. 40
    4. 10
    Answer

    D. 10

    60 + 50 − 100 = 10.

31. Consider the statements: 1. The empty set is a subset of every set. 2. The empty set has exactly one element.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    ϕ is a subset of every set, but it has no element.

32. Consider the statements: 1. A − B and B − A are always equal. 2. A − B = A ∩ B′.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    A − B and B − A differ in general, but A − B = A ∩ B′ is true.

33. Consider the statements: 1. (A ∩ B)′ = A′ ∪ B′. 2. (A ∪ B)′ = A′ ∪ B′.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Only the first is a correct form of De Morgan's law.

34. Consider the statements: 1. A set with n elements has 2^n subsets. 2. A set with n elements has 2^n − 1 proper subsets.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    All subsets are 2^n; excluding the set itself gives 2^n − 1.

35. Consider the statements: 1. {1, 2, 2, 3} has four elements. 2. {1, 2, 3} and {3, 2, 1} are the same set.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Repeated elements are counted once, and order does not matter.

36. Consider the statements: 1. n(A ∪ B) = n(A) + n(B) always. 2. n(A ∪ B) = n(A) + n(B) when A and B are disjoint.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    The simple sum works only when there is no overlap.

37. Consider the statements: 1. {0} is the empty set. 2. n({ϕ}) = 1.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    {0} has one element; {ϕ} also has exactly one element.

38. Consider the statements: 1. A ∪ A′ = U. 2. A ∩ A′ = ϕ.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    These are the two complement laws.

39. Consider the statements: 1. Equal sets are always equivalent. 2. Equivalent sets are always equal.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Equal sets have the same count; {1, 2} and {3, 4} are equivalent but not equal.

40. Consider the statements: 1. The set of tall students of a class is a well-defined set. 2. The set of students taller than 150 cm in a class is a well-defined set.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    'Tall' is vague; '150 cm' is a clear rule.

41. Consider the statements: 1. If A ⊂ B and B ⊂ A, then A = B. 2. Every set is a proper subset of itself.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A set is a subset of itself, but not a proper subset.

42. Consider the statements: 1. Venn diagrams use closed curves to represent sets. 2. Overlapping circles in a Venn diagram show that the sets have common elements.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard descriptions.

43. Consider the statements: 1. 2 ⊂ {1, 2, 3}. 2. {2} ⊂ {1, 2, 3}.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    A subset symbol joins two sets; 2 is an element.