Review for Test 1

Review for Test 1 - Is the work correct Is the answer...

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1) Let p and q be statements. Are ( p ∨ ∼ q ) ( p ∧ ∼ q ) and p equivalent statement forms? Prove or disprove according to your answer. 2) Let D = {- 1 , 0 , 1 } . Consider the following statement: n D, m D such that n < m 2 a) Is the statement you wrote in (b) true? Explain. b) Write the negation of the statement. c) Is the statement you wrote in (b) true? Explain. 3) Let p , q , r , s and t be statements. Prove that the following argument is valid: p q -→ r s ∨ ∼ q t p -→ t p r -→ ∼ s q 4) Let p and q be statements. a) Circle all the statements below that are equivalent to p q . (No explanations needed.) i) ( p ) ( q ) ii) ( q ) ( p ) iii) ( p ) q iv) p ( q ) v) q p
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2 b) Circle all the statements below that are equivalent to ( p q ). (No explana- tions needed.) i) ( p ) ( q ) ii) ( q ) ( p ) iii) ( p ) q iv) p ( q ) v) q p 5) A student wants to solve the following equation: x 2 x 2 - 1 = 4 x - 3 x 2 - 1 Here is the work: x 2 x 2 - 1 = 4 x - 3 x 2 - 1 x 2 = 4 x - 3 x 2 - 4 x + 3 = 0 ( x - 1)( x - 3) = 0 x = 1 or x = 3 Answer: the solution set consists of the real numbers 1 and 3.
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Unformatted text preview: Is the work correct? Is the answer correct? Explain. 6) Let p , q and r be statements. Are ( p ∧ q ) → r and ( ∼ p ) ∨ ( ∼ q ) ∨ r equivalent statement forms? Prove or disprove according to your answer. 7) Prove the following universal statement: ∀ integers n and m , if 7 | ( n-m ) then 7 | ( n 3-m 3 ). 3 8) Let p , q and r be statements. Is the following argument valid? Prove or disprove according to your answer. p → ( q ∨ r ) r → q ∴ q 9) Let p and q be statements. Circle all the statements below that are equivalent to p → q . (No explanations needed.) a) ( ∼ p ) → ( ∼ q ) b) ( ∼ q ) → ( ∼ p ) c) ( ∼ p ) ∨ q d) p ∨ ( ∼ q ) e) q → p 10) Prove: “ ∀ integer n ≥ 0: if 5 | n then the decimal representation of n ends in 0 or in 5 ”...
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This test prep was uploaded on 04/15/2008 for the course M 324K taught by Professor Sofer during the Spring '08 term at University of Texas.

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Review for Test 1 - Is the work correct Is the answer...

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