17 R.2 Sets and Real Numbers Every element of the set S = 51, 2, 3, 46 is a natural number. S is an example of a subset of the set N of natural numbers. This relationship is written using the symbol ⊆. S⊆N By definition, set A is a subset of set B if every element of set A is also an element of set B. For example, if A = 52, 5, 96 and B = 52, 3, 5, 6, 9, 106, then A⊆B. However, there are some elements of B that are not in A, so B is not a subset of A. This relationship is written using the symbol h. BsA Every set is a subset of itself. Also, ∅ is a subset of every set. If A is any set, then A#A and ∅#A. Figure 8 shows a set A that is a subset of set B. The rectangle in the drawing represents the universal set U. Such a diagram is a Venn diagram. Two sets A and B are equal whenever A⊆B and B⊆A. Equivalently, A = B if the two sets contain exactly the same elements. For example, 51, 2, 36 = 53, 1, 26 is true because both sets contain exactly the same elements. However, 51, 2, 36≠50, 1, 2, 36 because the set 50, 1, 2, 36 contains the element 0, which is not an element of 51, 2, 36. U B A A # B Figure 8 EXAMPLE 3 Examining Subset Relationships Let U = 51, 3, 5, 7, 9, 11, 136, A = 51, 3, 5, 7, 9, 116, B = 51, 3, 7, 96, C=53, 9, 116, and D=51, 96. Determine whether each statement is true or false. (a) D⊆B (b) B⊆D (c) CsA (d) U= A SOLUTION (a) All elements of D, namely 1 and 9, are also elements of B, so D is a subset of B, and D⊆B is true. (b) There is at least one element of B (for example, 3) that is not an element of D, so B is not a subset of D. Thus, B⊆D is false. (c) C is a subset of A, because every element of C is also an element of A. Thus, C⊆A is true, and as a result, CsA is false. (d) U contains the element 13, but A does not. Therefore, U= A is false. S Now Try Exercises 53, 55, 63, and 65. Operations on Sets Given a set A and a universal set U, the set of all elements of U that do not belong to set A is the complement of set A. For example, if set A is the set of all students in a class 30 years old or older, and set U is the set of all students in the class, then the complement of A would be the set of all students in the class younger than age 30. The complement of set A is written A′ (read “A-prime”). The Venn diagram in Figure 9 shows a set A. Its complement, A′, is in color. Using set-builder notation, the complement of set A is described as follows. A′ = 5x∣ x{U, xoA6 A A9 A9 Figure 9
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