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assig2 - y and z(b There is no smallest positive real...

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MATH 135 Algebra, Assignment 2 Due: Wed Sept 30, 8:30 am Please note that the symbols ¬ , , , and are alternate notations for the connectives “NOT”, “AND”, “OR”, “= ” and “ ⇐⇒ ” respectively. 1: (a) Make a truth table for the statement ( P ↔ ¬ R ) ( R Q ). (b) Determine whether ( P ∧ ¬ Q ) ( R P ) is equivalent to Q R . (c) Determine whether P ( Q R ) is equivalent to ( P Q ) R . 2: Let S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . List all of the elements in each of the following sets. (a) A = x S x is even or x is a multiple of 3 (b) B = x S if x is even then x is a multiple of 3 (c) C = ( x, y ) S × S 3 x + 2 y = 20 and ( y < x 2 if and only if xy < 8) 3: Express each of the statements below symbolically, taking the universe of discourse to be R and using only symbols from the following list: ¬ , , , , , ( , ) , , , 0 , 1 , + , × , = , < , , x , y , z (a) x is equal to the minimum of
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Unformatted text preview: y and z . (b) There is no smallest positive real number. (c) Every real number has a unique cubed root. 4: For each of the following statements, determine whether it is true when the universe of dis-course is Z and whether it is true when the universe of discourse is R . (a) ∀ x ∃ y ( x = y × y ∨ y × y + x = 0) (b) ∀ x ∃ y ( y < x ∧ ∀ z ( z < x → z ≤ y ) ) (c) ∀ x ∀ y ( (0 < x ∧ x + y < x × y ) → < y ) 5: Prove each of the following statements, where the universe of discourse is R . (a) ∀ x ( | x 2-4 x | ≤ x ↔ ( x = 0 ∨ (3 ≤ x ∧ x ≤ 5) )) (b) ∀ x ∀ y ( y 2 = x 3 + x 2 ↔ ∃ z ( x = z 2-1 ∧ y = z 3-z ) )...
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