(b) Show the following. Two mappings f:X→ Y and g:X→ Y are said to be equal (and we write this f = g) if f(x) = g(x) for every x in X. Let f, g, and h be any three mappings of a non-empty set X into itself, and show that multiplication of mappings is associative in the sense that f(gh) = (fg)h.

Q: Consider the functions f : A → B, g : B → A and h : B → A such that if gf = 1A and fh = 1B. Prove…

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Q: A = {1,2,3}, B = {4,5,6}, C = {7,8,9,10} define the functions f :A→C and g:B→C by f(1) = 7, f(2) =…

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Q: Consider the relation R = {(a, b) ∈ Z × Z : 3a − 5b is even }. Show that R is an equivalence…

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Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A → A and ƒ: A → A. (a)…

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Q: Disprove that for any relations R and S, ran(S • R) = ran(S).

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Q: An ordering function is a function f:Z→Z which takes the set of integers, and maps them to another…

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Q: Calculus Question

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Q: Let f: A --> B and g: B --> C be maps. Show that if g of f is 1-1 and onto, then f is 1-1 and g is…

A: According to the given information,

Q: Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and…

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Q: Let A = {1,2,3}, B B = = Define the functions f: AC and g: B → C by f(1) = 7, f(2)= 9, f(3) = 7,…

A: We want g o h = f.     So, we want g o h(1) = f(1) = 7            and g o h(3) = f(3) = 7   Now,…

Q: For each of the following mappings f: R---------- R, determine whether the mapping is onto and…

A: Given function f(x) = x     first we check that function is one - one or not                take…

Q: (i) Let A = {1,2,3,4,5), then either find a relation on A that is not reflexive on A and not…

A: Solution:

Q: (a) Give an example of two functions f: A→B and g: B-C such that f is both 1-1 and onto but gof is…

A: (a)let's define sets 𝐴, 𝐵, and 𝐶 as follows:Let 𝐴={1,2}Let 𝐵={3,4}Let 𝐶={5}Now, let's define…

Q: Prove the following: Theorem. Assume that f: A → B and g: B → A are injective maps. Then there…

A: Given:f: A→B     injective mapg:B→A     injective mapTo Proof:There exist bijective map h: A→B.…

Q: For a set A, let {0,1} A be the set of all functions from A to the two point set {0,1}. Prove that…

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Q: Let R be the relation on N defined by x Ry if x and y share a common factor other than 1. Determine…

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Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

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Q: Let E = {1, 2, 3} and F = {-2, –1, 0} and define a relation T from E to F as follows: For every (х,…

A: Since you have posted multipart question and we can answer three subparts. So, answering the first…

Q: a) Give an example of an algebraic function f : Z → Z that is injective but not surjective. You do…

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Q: A relation R on R² is defined by ((x1, y1), (2, 2)) R if Prove that R is an equivalence relation.…

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Q: Define a relation R on the x-y plane as follows: Given two points (x1 ,y1) and (x2 ,y2) in the x-y…

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Q: Let X and Y be Banach spaces and F:X → Y be a linear map which is continuous and open. Will F always…

A: The answer provided below has been developed in a clear step by step manner.

Q: Let A = (0,1,2,3). The relations R, S and T in the set A are defined as follows……

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Q: Let ℝ denote the set of real numbers, and  β:={(x,y)∈ℝxℝ| xy≥∅} be a relation on ℝ. Which…

A: The given relation β⊂ℝ2 is: β=x,y∈ℝ×ℝ | xy≥0 The properties of a relation is: Reflexive Irreflexive…