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1. Apply a truth table to show each conclusion of following: (a) ~(~p) = p (b)-(pv q) = (~p) ^ (~q)

1. Apply a truth table to show each conclusion of following: (a) ~(~p) = p (b)-(pv q) = (~p) ^ (~q)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Apply a truth table to show each conclusion of following: (a) ~(~p) = p (b) ~(pVq) = (~p) ^ (~q) 2. Write down the converse, inverse and contra-positive of each of the following statements: (a) For any real number x, if x >4, then x² > 16. (b) If both a and b are integers, then their product ab is an integer. 3. Use logical equivalences to simplify each one of following a) ((PA¬Q) V (PAQ)) ^Q (b)-((-p^q) v (p^-q)) v (p^q) 4. Negating the following statements: (a) V primes p, p is odd. (b) 3 a triangle T such that the sum of the angles equals 200°. (c) For every square x there is a triangle y such that x and y have different colors. (d) There exists a triangle y such that for every square x, x and y have different colors. (e) V people p, if p is blond then p has blue eyes. 5. Construct a truth table to determine whether or not the argument is valid (a) (b) pv (q vr) יזר pv q p→qv (¬r) q→ p^r :p →r 6. Prove that (a) 9n² + 3n-2 is even for any integer n. (b) For all integers mann, m+nand m-nare either both odd or both even. (c) There are real numbers such that √a + b =√a + √b. (d) For all integers, if n is odd then n²is odd. 7. Show that the following statements are false: (a) There is an integer n such that 2n² - 5n + 2 is a prime. (b) If m and n are any two positive integers then mn > m + n.
Transcribed Image Text:1. Apply a truth table to show each conclusion of following: (a) ~(~p) = p (b) ~(pVq) = (~p) ^ (~q) 2. Write down the converse, inverse and contra-positive of each of the following statements: (a) For any real number x, if x >4, then x² > 16. (b) If both a and b are integers, then their product ab is an integer. 3. Use logical equivalences to simplify each one of following a) ((PA¬Q) V (PAQ)) ^Q (b)-((-p^q) v (p^-q)) v (p^q) 4. Negating the following statements: (a) V primes p, p is odd. (b) 3 a triangle T such that the sum of the angles equals 200°. (c) For every square x there is a triangle y such that x and y have different colors. (d) There exists a triangle y such that for every square x, x and y have different colors. (e) V people p, if p is blond then p has blue eyes. 5. Construct a truth table to determine whether or not the argument is valid (a) (b) pv (q vr) יזר pv q p→qv (¬r) q→ p^r :p →r 6. Prove that (a) 9n² + 3n-2 is even for any integer n. (b) For all integers mann, m+nand m-nare either both odd or both even. (c) There are real numbers such that √a + b =√a + √b. (d) For all integers, if n is odd then n²is odd. 7. Show that the following statements are false: (a) There is an integer n such that 2n² - 5n + 2 is a prime. (b) If m and n are any two positive integers then mn > m + n.
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