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CSE 260
QUIZ5– Proofs  ANSWER
(20 minutes)
NAME:
1. Indicate if the following are trivially true or vacuously true for a real number x and
n=1.
(a) If
x
≥
0, then (1 +
x
)
n
≥
1 +
nx
Trivially true (
p
→
q
, the implication is trivially true if q is true; in this case, the
truth value of p is irrelevant).
(b) If n is even, then
x
n
≥
0
Vacuously true (
p
→
q
, the implication is vacuously true if
p
is false).
2. Prove that if n is an integer and 3n+2 is odd, then n is odd using
(a) an indirect proof.
Theorem: (3n+2 is odd)
→
(n is odd)
Contrapositive:
¬
(n is odd )
→ ¬
(3n+2 is odd)
⇔
(n is even)
→
(3n+2 is even)
n is even. Therefore, by the mathematical deFnition of even n can be written as
n=2k for an integer k
Therefore, 3n+2=3(2k)+2=2(3k+1) is even becuase 3k+1 is an integer
(b) A proof by contradiction.
Theorem: (3n+2 is odd)
→
(n is odd)
Assume that the theorem is false, i.e., for all integers n
¬
((3n+2 is odd )
→
(n is odd))
⇔ ¬
(
¬
(3n+2 is odd)
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This note was uploaded on 03/30/2008 for the course CSE 260 taught by Professor Saktipramanik during the Spring '08 term at Michigan State University.
 Spring '08
 SaktiPramanik

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