discrete_mathematics_test_bank

In 76–77, express the negation of the statement in terms of quantifiers without using the negation symbol. 76. ∀x((x −1) ∨ (x 1)) 77. ∃x(3 x ≤ 7) In 78–79 P (x, y) means “x and y are real numbers such that x + 2y = 5 .” Determine whether the statement is true. 78. ∀x∃y P (x, y) 79. ∃x∀y P (x, y) In 80–82 P (m, n) means “m ≤ n ,” where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? 80. ∀n P (0, n) 81. ∃n∀m P (m, n) 82. ∀m∃n P (m, n) In questions 83–88 suppose P (x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose P (1, 3), P (2, 1), P (2, 2), P (2, 3), P (3, 1), P (3, 2) are true, and P (x, y) is false otherwise. Determine whether the following statements are true. 83. ∀x∃yP (x, y) 84. ∃x∀yP (x, y) 85. ¬∃x∃y (P (x, y) ∧ ¬P (y, x)) 86. ∀y∃x (P (x, y) → P (y, x)) 87. ∀x∀y (x /= y → (P (x, y) ∨ P (y, x)) 88. ∀y∃x (x ≤ y ∧ P (x, y)) In 88–92 suppose the variable x represents students and y represents courses, and: U (y): y is an upper-level course M (y): y is a math course F (x): x is a freshman B(x): x is a full-time student T (x, y): student x is taking course y . Write the statement using these predicates and any needed quantifiers. 89. Eric is taking MTH 281. 90. All students are freshmen. 91. Every freshman is a full-time student. 92. No math course is upper-level. In 93–95 suppose the variable x represents students and y represents courses, and: U (y): y is an upper-level course M (y): y is a math course F (x): x is a freshman A(x): x is a part-time student T (x, y): student x is taking course y . Write the statement using these predicates and any needed quantifiers. 93. Every student is taking at least one course. 94. There is a part-time student who is not taking any math course. 95. Every part-time freshman is taking some upper-level course. In 96–98 suppose the variable x represents students and y represents courses, and: F (x): x is a freshman A(x): x is a part-time student T (x, y): x is taking y . Write the statement in good English without using variables in your answers. 96. F (Mikko) 97. ¬∃y T (Joe, y) 98. ∃x (A(x) ∧ ¬F (x)) In 99–101 suppose the variable x represents students and y represents courses, and: M (y): y is a math course F (x): x is a freshman B(x): x is a full-time student T (x, y): x is taking y . Write the statement in good English without using variables in your answers. 99. ∀x∃y T (x, y) 100. ∃x∀y T (x, y) 101. ∀x∃y [(B(x) ∧ F (x)) → (M (y) ∧ T (x, y))] In 102–104 suppose the variables x and y represent real numbers, and L(x, y) : x y G(x) : x 0 P (x) : x is a prime number. Write the statement in good English without using any variables in your answer. 102. L(7, 3) 103. ∀x∃y L(x, y) 104. ∀x∃y [G(x) → (P (y) ∧ L(x, y))] In 105–107 suppose the variables x and y represent real numbers, and L(x, y) : x y Q(x, y) : x = y E(x) : x is even I(x) : x is an integer.