Question

Very clear and step by step proof would be very appreciated. Thanks in advance.

Consider the set X, and let F = {f|j: X rightarrow X), the set of functions from X to itself. A notable member of F is, of course, the identity function, id_x: X rightarrow X, defined by the formula: Forall x Element X. id_x(x) = x. In this exercise, we will study the concept of the inverse function. Recall that given two functions f, g Element F we have defined the operation of composition, which is denoted o and takes two functions f, g Element F and produces a new function as follows: f og(x) = f(g(x) for all x Element X. We will give the following definitions: (i) Def 1. A function g Element F if called an left inverse of a function f Element F go f = id_x. (ii) Def 2. A function h Element F if called an right inverse of a function f Element F if f o h = id_x. (iii) Def 3. A function l Element F if called an inverse of a function f Element F if l is simultaneously a left and a right inverse of f. Let LI(f) the predicate "f has a left inverse", RI(f) the predicate "f has a right inverse", and INV(f) the predicate "f has an inverse". Also let INJ(f) be the predicate "f is injective (one-to-one)", SUR(f) "f is surjective (onto)", and "BIJ(f) "f is bijective (one-to-one and onto). Express each of the following statements in predicate logic and write a proof. (i) A function f: X rightarrow X is injective if and only if it has a left inverse. (ii) A function f: X rightarrow X is surjective if and only if it has a right inverse. (iii) A function f: X rightarrow X is bijective if and only if it has an inverse. (iv) If the inverse g of a function f: X rightarrow X exists, it is unique. ("Unique" means the following: if f has another inverse g', then g = g'. Since the inverse of a function is unique, there is a special notation used for it: f ^-1).

Show transcribed image text Consider the set X, and let F = {f|j: X rightarrow X), the set of functions from X to itself. A notable member of F is, of course, the identity function, id_x: X rightarrow X, defined by the formula: Forall x Element X. id_x(x) = x. In this exercise, we will study the concept of the inverse function. Recall that given two functions f, g Element F we have defined the operation of composition, which is denoted o and takes two functions f, g Element F and produces a new function as follows: f og(x) = f(g(x) for all x Element X. We will give the following definitions: (i) Def 1. A function g Element F if called an left inverse of a function f Element F go f = id_x. (ii) Def 2. A function h Element F if called an right inverse of a function f Element F if f o h = id_x. (iii) Def 3. A function l Element F if called an inverse of a function f Element F if l is simultaneously a left and a right inverse of f. Let LI(f) the predicate "f has a left inverse", RI(f) the predicate "f has a right inverse", and INV(f) the predicate "f has an inverse". Also let INJ(f) be the predicate "f is injective (one-to-one)", SUR(f) "f is surjective (onto)", and "BIJ(f) "f is bijective (one-to-one and onto). Express each of the following statements in predicate logic and write a proof. (i) A function f: X rightarrow X is injective if and only if it has a left inverse. (ii) A function f: X rightarrow X is surjective if and only if it has a right inverse. (iii) A function f: X rightarrow X is bijective if and only if it has an inverse. (iv) If the inverse g of a function f: X rightarrow X exists, it is unique. ("Unique" means the following: if f has another inverse g', then g = g'. Since the inverse of a function is unique, there is a special notation used for it: f ^-1).