222a1 - A . Prove or give a counterexample to each of the...

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MATH 222 FALL 2011, Assignment One (due Thursday, Sep. 22 in class before the lecture begins) Show yourwork clearly. Illegible or disorganized solutions will receive no credit. 1. Let f : N N be deFned by f = { ( x, 3 x ) : x N } . Let S be the set of all even natural numbers. ±ind the following sets. (a) f ( S ). [1] (b) f - 1 ( S ). [1] (c) f - 1 (2011). [1] (d) S f ( N ). [1] 2. Determine all x R such that x + x + 1 2 = 2 x . [3] 3. Given an example of a function from N to N that is (a) one-to-one but not onto; [2] (b) onto but not one-to-one; [2] (c) both onto and one-to-one (but di²erent from the identity function); [2] (d) neither onto nor one-to-one. [2] 4. Suppose that f is a function from A to B and g is a function from C to D . (a) Is f g a function from A C to B D ? Explain! [2] (b) Construct such an example of f and g with f n = g that f g is a function from A C to B D . [2] 5. Suppose that R 1 and R 2 are relations on a set
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Unformatted text preview: A . Prove or give a counterexample to each of the following assertions. (a) If R 1 and R 2 are symmetric, then R 1 R 2 is symmetric. [2] (b) If R 1 and R 2 are antisymmetric, then R 1 R 2 is antisymmetric. [2] (c) If R 1 R 2 is reexive, then either R 1 or R 2 is reexive. [2] (d) If R 1 R 2 is reexive, then both R 1 and R 2 are reexive. [2] 6. If A = { a, b, c, d, e, f, g } , determine the number of relations on A that are (a) reexive and symmetric. [2] (b) rexive and contain ( a, b ) and ( b, c ). [2] 7. Let R consists of all pairs ( x, y ) Z Z such that x 2 y 2 is divisible by 3. (a) Show that R is an equivalence relation. [3] (b) Determine the partition induced by R . [2] 1...
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