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4. Determine whether each of the functions below is onto, and/or one-to-one for f : Z → Z, prove your answers. (a) f(x) = 5x − 3 (b) f(x) = 2x 3 (c) f(x) = (2x − 2)2 (d) f(x) = √ x 6 5. Determine whether each of the functions below is onto, and/or one-to-one for f : R → R, prove your answers. (a) f(x) = 5x − 3 (b) f(x) = 2x 3 (c) f(x) = (2x − 2)2 (d) f(x) = √ x 6 6. Define the following functions (assume that the domains/codomains are defined such that each composition is valid): f(x) = 2x, g(x) = x (1+x) , h(x) = √ x. Find (a) f ◦ g ◦ h (b) h ◦ g ◦ f(c) f ◦ f (d) g ◦ g 7. Find inverses of the following functions (assume that the domains/codomains are defined such that each function is a bijection). (a) f(x) = 5x − 3 (b) f(x) = 2x 3 (c) f(x) = (2x − 2)2 (d) f(x) = √ x 8. 6 Let f(x) and g(x) be two linear functions (a function is linear if f(x) = ax + b for a, b ∈ R). a. Prove or disprove: f ◦ g = g ◦ f. b. Prove or disprove: f ◦ g and g ◦ f are linear functions. 9. Let R be a relation on a set A = {a1, . . . , an} of size n. Let MR be the 0-1 matrix representing R (i.e., the entry mij = 1 if (ai , aj ) ∈ R and zero otherwise). (a) How many unique relations are there on A (in terms of n)? (b) The complement relation is defined as R = {(a, b) | (a, b) 6∈ R} Say that the number of nonzero entries in MR (that is, the number of 1s) is k. How many nonzero entries are there in MR ? Briefly justify your answer. (c) How many reflexive relations are there on a set of size n? Briefly justify your answer. (d) How many symmetric relations are there on a set of size n? Briefly justify your answer. 10. Define the following relation on the set of integers. R1 = {(a, b) | a > b, a, b ∈ Z} (a) Give an element that is in R1 (b) Give an element that is not in R1. (c) Give an element that is in R1 ◦ R1 (d) Give an element that is not in R1 ◦ R1. (e) Give an element that is in R3 1 (f) Give an element that is not in R3 1 (g) Give a general (set builder) definition for Rn 1 in terms of a, b and n. 11. Prove the following. A relation R is asymmetric if and only if R is irreflexive and antisymmet-ric. Note: a relation R on the set A is irreflexive if for every a ∈ A, (a, a) 6∈ R. That is, R is irreflexive if no element in A is related to itself. Hint: write the definition of what it means to be asymmetric, then “add” the contradiction:

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