23 R.2 Sets and Real Numbers Insert ⊆ or sin each blank to make the resulting statement true. See Example 3. 63. 52, 4, 66 52, 3, 4, 5, 66 64. 51, 56 50, 1, 2, 3, 56 65. 50, 1, 26 51, 2, 3, 4, 56 66. 55, 6, 7, 86 51, 2, 3, 4, 5, 6, 76 67. ∅ 51, 4, 6, 86 68. ∅ ∅ Determine whether each statement is true or false. See Examples 4–6. 69. 55, 7, 9, 196¨57, 9, 11, 156 = 57, 96 70. 58, 11, 156¨58, 11, 19, 206 = 58, 116 71. 51, 2, 76´51, 5, 96 = 516 72. 56, 12, 14, 166´56, 14, 196 = 56, 146 73. 52, 3, 5, 96¨52, 7, 8, 106 = 526 74. 56, 8, 96´59, 8, 66 = 58, 96 75. 53, 5, 9, 106¨ ∅= 53, 5, 9, 106 76. 53, 5, 9, 106´∅= 53, 5, 9, 106 77. 51, 2, 46´51, 2, 46 = 51, 2, 46 78. 51, 2, 46¨51, 2, 46 = ∅ 79. ∅´∅= ∅ 80. ∅¨∅= ∅ Let U= 50, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 136, M= 50, 2, 4, 6, 86, N = 51, 3, 5, 7, 9, 11, 136, Q= 50, 2, 4, 6, 8, 10, 126, and R = 50, 1, 2, 3, 46. Use these sets to find each of the following. Identify any disjoint sets. See Examples 4–6. 81. M¨R 82. M¨U 83. M´N 84. M´R 85. M¨N 86. U¨N 87. N´R 88. M´Q 89. N′ 90. Q′ 91. M′ ¨Q 92. Q¨R′ 93. ∅¨R 94. ∅¨Q 95. N´∅ 96. R´∅ 97. 1M¨N2 ´R 98. 1N´R2 ¨M 99. 1Q¨M2 ´R 100. 1R´N2 ¨M′ 101. 1M′ ´Q2 ¨R 102. Q¨1M´N2 103. Q′ ¨1N′ ¨U2 104. 1U¨∅′2 ´R 105. 5x0 x ∈U, x ∉M6 106. 5x0 x ∈U, x ∉R6 107. 5x0 x ∈M and x ∈Q6 108. 5x0 x ∈Q and x ∈R6 109. 5x0 x ∈M or x ∈Q6 110. 5x0 x ∈Q or x ∈R6 Let A = E -6, - 12 4 , - 5 8 , - 23, 0, 1 4 , 1, 2p, 3, 212 F. List all the elements of A that belong to each set. See Example 7. 111. Natural numbers 112. Whole numbers 113. Integers 114. Rational numbers 115. Irrational numbers 116. Real numbers
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