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Unformatted text preview: Homework 5
CS 2311
Due: 26Sep03 (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.
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 Fall '08
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