Unformatted text preview: q and squaring both sides we have 3 q 2 = p 2 . Thus, p 2 is divisible by 3. So p is divisible by 3. Substitute p = 3 k in to the equation we get 3 q 2 = 9 k 2 , or equivalently q 2 = 3 k 2 which implies q is divisible by 3. This contradicts our assumption that p,q have no common factors. Thus √ 3 is irrational. b) We will prove x is rational iﬀ y is rational, which is equivalent to the given statement. Assume x is rational, then x = p q (for integers p,q , q 6 = 0). But y = 4-x = 4 q-p q . Thus y is also rational. The opposite direction that y is rational implies x is rational can be proved similarly. c) x = 0 ,y = √ 2 is a counterexample. 1...
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- Winter '07
- Fraction, Prime number, Rational number, Integral domain, PROBLEM PROBLEM PROBLEM