Let f A → B and g : B → C be functions. For each of the following,prove or provide a counterexample.(a) If go f is injective, is f always injective?(b) Ifgof is injective, is g always injective?(c) If go f is surjective, is f always surjective?(d) If g of is surjective, is always surjective? For this homework, you can use the following facts. If it is a definition, youshould mention "by definition" or "by definition of XXX (e.g. even)". If it isnot a definition, you do NOT have to cite.1. Basic algebra, such as 2 + 2 = 4, 1-3 = -2, 2-4 = 8. This includessubtracting an integer, or dividing by a non-zero rational/real/complexnumbers.2. Common knowledge of whether numbers are integer/rational, e.g. is notan integer, is not rational, all integers are rational, all rational numbersare real, all real numbers are complex, etc.3. Associative law of addition: (a+b)+c= a + (b+c), and associativelaw of multiplication: (ab)c = a(be), for all a, b, c € C.4. Commutative law of addition: a+b=b+a, and commutative lawof multiplication: abba, for all a, b, c € C.5. Distributive law: For all a, b, c € C, we have (a + b)(c+d) = ac+ad+be + bd. In particular, (a + b)² = a² + 2ab + b².6. An integer n is even if there exists an integer k such that n = 2k.7. An integer n is odd if there exists an integer k such that n = 2k + 1.8. All integers are either even or odd.9. A real number az is positive if a > 0, and negative if x < 0.10. All real numbers are either positive, negative, or 0.11. Let A, B be subsets of a universal set U.(a) We say A is a subset of B, denoted by ACB, if for all z € A, wehave z € B.(b) An element is in AUB, the union of A and B, if z € A or € B.(c) An element z is in An B, the intersection of A and B, if x € A and€ B.(d) An element z EU is in Aº, the complement of A in U, if æ & A.12. We denote by the empty set.13. Let A, B be sets. Then(a) An BCA;(b) ACAUB.114. A "function" f: A → B is a well-defined function, or simply a function,if for all z EA, there is a unique y E B such that f(x) = y. (There arequotes here as f is not actually a function if it is not well-defined. )15. A function f: A → B is injective if for all ₁,22 € A, if f(x₁) = f(x₂),then #1 = 22-16. A function f : A → B is surjective if for all y € B, there exists € A suchthat f(x) = y.17. A function f : A → B is bijective if it is injective and surjective.

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Let f A → B and g : B → C be functions. For each of the following, prove or provide a counterexample. (a) If 9 of is injective, is f always injective? (b) If go f is injective, is g always injective? (c) If 9 of is surjective, is ƒ always surjective? (d) If go f is surjective, is g always surjective?