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# 2 - Math 135 Assignment 2 Solutions Winter 2009 1 Express...

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Math 135 Assignment 2 Solutions Winter 2009 1. Express each statement as a logical expression using quantifiers. State the universe of discourse. (a) There is a smallest positive integer. a b ”0 < a and a b ”, UD is the integers. (b) There is no smallest positive real number. Not a b a b ”, UD is the real numbers. (c) Every integer is the sum of two integers. a, b c a = b + c ”, UD is the integers. (d) Every pair of integers has a common divisor. a, b, c, d, e a = cd and b = ce ”, UD is the integers. (e) There is a real number x such that, for every real number y , y 2 - y = x . x, y y 2 - y = x ”, UD is the real numbers. (f) For every real number y , there is a real number x such that x 3 + x = y . y, x x 3 + x = y ”, UD is the real numbers. (g) The equation x 3 + y 3 = z 3 has no positive integer solutions. Not a, b, c ”0 < c and 0 < b and 0 < c and a 3 + b 3 = c 3 ”, UD is the integers. 2. Write down the converse and contrapositive of each statement.

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2 - Math 135 Assignment 2 Solutions Winter 2009 1 Express...

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