MAT-2510 Midterm: MATH 2510 App State Spring2015 Test2
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Name: (30 points) converging questions (a) prove that (cid:28) 2n3 + n n3 + 3 (cid:29) converges. Show that anbn 0: (20 points) some set stu . (a) let a, b, c be sets. Show a (b c) = (a b) (a c). (b) let f : x y be a function and t1, t2 y . Show that f 1(t1 t2) = f 1(t1) f 1(t2). (c) let f : x y and let r1, r2 x. Prove that if f is one-to-one, then f (r1 r2) = f (r1) f (r2). Half of your proof does not need the one-to-one hypothesis: (25 points) for each of the following functions, decide if f is 1-1, onto, both, or neither. 2 x is even x is odd (d) show that composing two one-to-one functions yields a one-to-one function and that composing two onto functions yields and onto function. Speci cally, let f : x y and g : y z.
