quebec - ∼ on Z as follows a ∼ b if and only if 3 a b...

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Math 210 Section 011: Test #1 NAME: This test has 6 pages . The points per page are 4, 4, 5, 5, 5, 5. [4 points] Question 1. Let P be the statement “ - 4 2 = 16.” Let Q be the statement “7 < 50.” Let R be the statement “ x = 7 is the only solution of ( x - 2) 2 = 25.” Give the truth values of each of the following. P R P Q Q R ( ¬ R ) ± P ( ¬ Q ) ² 1
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[2 points] Question 2a. Explicitly write out the power set of n a, { b } o . [2 points] Question 2b. Let A = { 1 , 2 , 4 , 5 , 6 , 9 } , let B = { 1 , 2 , 3 , 5 } , and let C = { 5 , 6 , 7 , 8 } . Explicitly write out the set ( A C ) × ( A B ). 2
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[3 points] Question 3a. Construct a truth table for ( P Q ) ± P ( ¬ Q ) ² . [2 points] Question 3b. If X Y is false, find the truth value of ± ( ¬ X ) Y ² ± ( ¬ X ) ( ¬ Y ) ² . 3
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[1 point] Question 4a. Show that the statement A ( B C ) = ( A B ) C is false by giving an explicit counterexample. [4 points] Question 4b. Classify each of the following as true or false. If true, give a general proof. If false, explicitly give a specific counterexample. (i) True or false: If A B = A C , then B = C . (ii) True or false: If A B = A C , then B = C . 4
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[5 points] Question 5. Define a relation
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Unformatted text preview: ∼ on Z as follows: a ∼ b if and only if 3 a + b is a multiple of 4. Prove that ∼ is an equivalence relation. 5 [5 points] Question 6. Suppose our universe of discourse is the set of all cats. Let S ( x ) be the predicate “ x loves salmon” Let T ( x ) be the predicate “ x is teachable” Let M ( x ) be the predicate “ x is missing a tail” Let W ( x ) be the predicate “ x has whiskers” Let G ( x ) be the predicate “ x has green eyes” Let K ( x ) be the predicate “Koko the gorilla likes to play with x ” Let F ( x,y ) be the predicate “ x is willing to fight with y .” Express each of the following as English sentences. ∃ x ( ¬ W ( x ) ) ∀ x ( M ( x ) → K ( x ) ) ∀ x ( G ( x ) → ∃ y F ( y,x ) ) ∀ x ± W ( x ) → ( S ( x ) → T ( x ) ) ² ∀ x ± S ( x ) → ∃ y ( ¬ F ( x,y ) ∧ ¬ F ( y,x ) ) ² 6...
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quebec - ∼ on Z as follows a ∼ b if and only if 3 a b...

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