5. Determine whether each of these functions from Z to Z is one-to-one (onto)a) f (n) = n − 1 b) f (n) = n2 + 1 c) f (n) = n3 d) 2nf n     6. Determine whether f : Z × Z → Z is onto ifa) f (m, n) = 2m − n b) f (m, n) = m2 − n2 c) f (m, n) = m + n + 1d) f (m, n) = |m| − |n| e) f (m, n) = m2 − 4 f) f (m, n) = m + n7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 18. Let S = {−1, 0, 2, 4, 7}. Find f (S) ifa) f (x) = 1 b) f (x) = 2x + 1 c) 5xf x     

4. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one (onto)a) f (a) = b, f (b) = a, f (c) = c, f (d) = db) f (a) = b, f (b) = b, f (c) = d, f (d) = cc) f (a) = d, f (b) = b, f (c) = c, f (d) = d

Which of the following function f: Z X Z → Z is not onto?a.f(a, B) = a – bb.f(a, B) = |b|c.f(a, B) = a + bd.f(a, B) = a

Prove.(a) f : Z+ → Z+, f (x) = 2x + 1 is one-to-one.(b) f : Z+ → Z, f (x) = 2x − x2 is not one to one.(c) f : R \ {0} → R+, f (x) = 1x2 is onto.(d) f : Z+ → Z+, f (x) = 2x − 1 is not onto.

For each of the following functions, state whether it is injective, surjective, and/or bijective, and why.(a) The function f (n) = 2n, mapping from integers to integers.(b) The function q(ϕ), with codomain N≥0, which maps any formula of predicate logic to the number of quantifiersin that formula

1. Why is f not a function from R to R ifa) f (x) = 1/x? b) f (x) =x ? c) f (x) =2 1x  ?2. Determine whether f is a function from Z to R ifa) f (n) = ±n b) 2 1f n n  c) 214f n n 3. Find these valuesa)1.1   b)0.1   c)4   d)3.2  e)5.2   e)2   e)1 22 3      4. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one (onto)a) f (a) = b, f (b) = a, f (c) = c, f (d) = db) f (a) = b, f (b) = b, f (c) = d, f (d) = cc) f (a) = d, f (b) = b, f (c) = c, f (d) = d5. Determine whether each of these functions from Z to Z is one-to-one (onto)a) f (n) = n − 1 b) f (n) = n2 + 1 c) f (n) = n3 d) 2nf n     6. Determine whether f : Z × Z → Z is onto ifa) f (m, n) = 2m − n b) f (m, n) = m2 − n2 c) f (m, n) = m + n + 1d) f (m, n) = |m| − |n| e) f (m, n) = m2 − 4 f) f (m, n) = m + n7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 18. Let S = {−1, 0, 2, 4, 7}. Find f (S) ifa) f (x) = 1 b) f (x) = 2x + 1 c) 5xf x     