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Homework5Solution - Homework 5 CS 2311 Due 26-Sep-03(Friday...

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Unformatted text preview: Homework 5 CS 2311 Due: 26-Sep-03 (Friday) 5:00pm [1) Match the term in the left column with the definition from the right col- 1.5 901 ' umn that best matches the word. Put the letter associated with a word’s 2? definition in the space next to the word. Q’— Axiom (a) A form of incorrect reasoning # Lemma (b) A proposition that is always true _QX._ Carollary {c} A prepsition that is always false __AL_ Theorem (d) A proposition that can be established directly from a theorem that has been proved _QL_ Fallacy (e) A statement accepted as true as the basis for argument or inference L Thutology (f) A simple theorem used in proof of another theorem + Centingency (g) A propostion that is neither always true nor always false # Contradiction (h) A statment that can be shown to be true (2) An argument is valid if whenever the premises 131,132, ..., P,1 are true, the conclusion Q is true. More formally, an argument is valid if the compound QFTS ‘ proposition: is a tautology. (3) Consider proof of the preposition p —> g. Match the proof technique in the left column with the phrase from the right column that best matches the ZFl'5 ‘ proof technique. Put the letter associated with the technique’s description in the space next to the technique. A Contradiction (a) Prove that p is always false LDirect (b) Prove (p1 —> g) {\{pg ———- {fin ...A(pn —r q), wherep=p1 sz V... Vpn & Vacuums (c) Prove -Iq —» --p _Q:.____ Trivial (d) Prove (p A -q) —» FALSE _L By Cas- (e) Prove that q is always true L Indirect (f) Assume p, derive q using known facts and rules of inference % Counterexample (g) Prove -u(p —> g] by finding a case in which 3) —r :1 does not hold /\ (4) Use a direct proof to show that the Sum of taro even integers is even. (5) Prove that if n is an integer and 3n+2 is even, then 11 is even, using proof by contraposition. (6) Prove that if n is an integer and 3n+2 is even, then :1 is even, using a / . (3 Fl} 125L911 proof by contradiction. (7) Prove that if n is a positive integer, then u is odd if and only if 711 + 4 is even. (Hint: Use the fact that p <-> q E (p —> (1) A (q —> 33).) .H {8) Use a proof by cases to show that min(a,min(b,c)) = min(min(o,b),c) \ whenever a, b, and c are real numbers. (9) Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. (10) Indicate the techniques used for each of the four proofs given on the fol- lowing pages. ‘31% . J F f 0 ‘ 22, Arm ‘ ILL—30 5 :: I :— M—H 22c a-z— + 3090? / ~—> - 9 ...
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