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SECTION 2.2 Subsets

SECTION 2.2 Subsets. The Universal Set The set of all elements that are being considered is called the universal set. We will use the letter U to denote the universal set. The Complement of a Set

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SECTION 2.2 Subsets

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  1. SECTION 2.2 Subsets The Universal Set The set of all elements that are being considered is called the universal set. We will use the letter U to denote the universal set. The Complement of a Set The complement of a set A, denoted by A', is the set of all elements of the universal set U that are not elements of A.

  2. EXAMPLE 1::: • Find the Complement of a Set Let U = {1,2,3,4,5,6,7,8,9,10}, S = {2,4,6,7}, and T = {x | x < 10 and x belongs to the odd counting numbers}. Find a. S' b. T'

  3. Solution::: a. The elements of the universal set are 1,2,3,4,5, 6,7,8,9, and 10. From these elements we wish to exclude the elements of S, which are 2, 4, 6, and 7. Therefore S' = {1, 3, 5, 8, 9, 10}. b. T = {1, 3, 5, 7, 9}. Excluding the elements of T from U gives us T' = {2, 4, 6, 8, 10}.

  4. CHECK YOUR PROGRESS Let U = {0, 2, 3, 4, 6, 7, 17}, M = {0, 4, 6, 17}, and P = {x | x < 7 and x is the even natural numbers}. Find a. M' b. P'

  5. Complement of the Universal Set and Complement of the Empty Set The Complement of the Universal Set is the Empty Set and the Complement of the Empty Set is Universal Set. i.e. U' = Øand Ø' = U.

  6. Properties of Subsets A Subset of a Set Definition Set A is a subset of set B, if and only if every element of A is also an element of B. • Here are two fundamental subset relationships. Subset Relationships 1) A is a subset of A 2) Ø is a subset of any set A

  7. EXAMPLE 2 • Determine whether each statement is true or false. a. {5, 10, I5, 20} is a subset of {0, 5, 10, 15, 20, 25, 30} b. W is subset of N c. {2, 4, 6} is a subset of {2, 4, 6} d. Ø is a subset of {1, 2, 3}

  8. Solution::: a. True; every element of the first set is an element of the second set. b. False; 0 is a whole number, but 0 is not a natural number. c. True; every set is a subset of itself. d. True; the empty set is a subset of every set.

  9. CHECK YOUR PROGRESS • Determine whether each statement is true or false. a. {1, 3, 5} is a subset of {1, 5, 9} b. The set of counting numbers is a subset of the set of natural numbers. c. { } is a subset of U. d. {-6, 0, 11} is a subset of I.

  10. Venn Diagram • The English logician John Venn (1834-1923) developed diagrams, which we now refer to as Venn diagrams, that can be used to illustrate sets and relationships between sets. In a Venn diagram, the universal set is represented by a rectangular region and subsets of the universal set are generally represented by oval or circular regions drawn inside the rectangle.

  11. Proper Subset Definition of Proper Subset Set A is a proper subset of set B, if every element of A is an element of B, and A is not equal to B. Venn diagrams can be used to represent proper subset relationships. For instance, if a set B is a proper subset of a set A, then we illustrate this relationship in a Venn diagram by drawing a circle labeled B inside of a circle labeled A.

  12. EXAMPLE • For each of the following, determine whether the first set is a proper subset of the second set. a. {a, e, i, o, u}, {e, i, o, u, a} b. N, I

  13. Solution::: a. Because the sets are equal, the first set is not a proper subset of the second set. b. Every natural number is an integer, so the set of natural numbers is a subset of the set of integers. The set of integers contains elements that are not natural numbers, such as - 3. Thus the set of natural numbers is a proper subset of the set of integers.

  14. CHECK YOUR PROGRESS • For each of the following, determine whether the first set is a proper subset of the second set. a. N, W b. {1, 4, 5}, {5, 1, 4}

  15. Subsets of a given Set EXAMPLE • List all the Subsets of a Set List all the subsets of {1, 2, 3, 4}.

  16. Solution::: An organized list produces the following subsets. • { } • {1}, {2}, {3}, {4} • {1, 2}, {l, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} • {1,2,3},{1,2,4},{1,3,4},{2,3,4}, • {1, 2, 3, 4}

  17. CHECK YOUR PROGRESS • List all of the subsets of {a, b, c, d, e}.

  18. The number of Subsets of a Set • The Number of Subsets of a Set • A set with n elements has 2n subsets.

  19. EXAMPLE::: • Find the number of subsets of each set. a. {1,2,3,4,5,6} b. {4, 5, 6, 7, 6, 5, 5}

  20. Solution::: a. {1, 2, 3, 4, 5, 6} has six elements. It has 26 = 64 subsets. b. {4, 5, 6, 7, 6, 5, 5} has 4 different elements. It has 24 =16 subsets.

  21. CHECK YOUR PROGRESS Find the number of subsets of the following set. {Mars, Jupiter, Pluto}

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