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hw04
Proof Strategies
Due 2007.Sep.20 (Thu) noon
Although you are welcome to typeset your work nicely (using Microsoft Word
or LaTeX or whatever to get nice logic symbols),
it's probably much easier to write formulas by hand.
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Rosen p85, #8
Prove that if n is a perfect square, then
n+2 is not a perfect square.
You'll certainly need a definition of “perfect square”;
see Rosen p.77, Example 2.
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Rosen p85, #11 variant
(= Rosen 5ed: p.75 #27 variant):
Prove or disprove that the sum of two irrational numbers
must be irrational.
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Rosen: p85, #12
(= Rosen 5ed: p.75 #28):
(Q-{0})·(ℜ-Q) ∈ (ℜ-Q)?
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Rosen: p85, #22
Show that if you pick three socks from a drawer having only
blue and black socks, you must get either a pair of blue socks
or a pair of black socks.
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Rosen: p102, #7 variant
(= Rosen 5ed: p.76 #49 variant):
Show that there are 5000 consecutive positive integers which are not
perfect squares.
A point of extra credit:
instead, show that for any non-negative integer n,
there are n consecutive positive integers which are not perfect squares.
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Rosen: p102, #10
(= Rosen 5ed: p.76 #52):
a pair with a nonnegative product
A point of extra credit:
instead, show that for any three real numbers a, b, c
the product of two of them is non-negative.
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Rosen: p102, #12:
Prove or disprove: a rational number raised to a rational number is rational.
“rationals closed under exponentiation?”
Hint: Recall that √2 = 21/2.
If you see a few other problems in Rosen which catch your eye,
and you'd like to do them for extra credit, you are welcome to
(though you can ask me for how much; extra-credit is harder to earn
point-per-point than regular credit).
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