Let R be the relation on R defined by R= {(x, y) E R × R | x² + y² = 4 or x = y²}. Answer each of the following parts by giving either a proof or a counterexample.

Q: 2 T= {(x, y): y = x2+4

A: The given relation,   T=x,y: y=2x2+4

Q: All the possible relations on A {1,2,3} is equal to i. i. 8 relations 9 relations Non of these iii.

A: The number of relations on a set of n elements is given by 2n2.

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

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Q: (b) Let R = {(x, y) E R × R : 1 < ¤ < 4 or x = 6}, and S = {(x, y) E R × R : -1 < y < 1}. i. Find…

A: Solution: The given relations are R=x,y∈ℝ×ℝ : 1≤x≤4 or x=6 and S=x,y∈ℝ×ℝ : -1&lt;y&lt;1

Q: b. Show that if f: A-> B is a bijection then |A| = |B| %3D c. Show that f:25 25 given by f(A) = S -…

A: Aa per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Exercise 1. Let R = {(x, y) | x, y E Z and x + y = 0, } be a relation on Z. Prove or disprove if R…

A: Let S be a non-empty set and R be a binary relation on S. (a) The relation R is said to be reflexive…

Q: 2. Let X1 and X2 be sets. Define functions T1 : X1 × X2 → X1, T2 : X1 × X2 → X2 by T| (X1, X2) = x1,…

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Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Let f : R → R be defined as f(x) = x3 + 8 (i) Prove that f is a bijection (one-to-one…

A: Consider the given function.

Q: (b) Let R= {(x,y) E R × R : 1 < x < 4 or x = 6}, %3D and S = {(x, y) E R × R: –1< y < 1}. i. Find…

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Q: Consider the relation S on N defined by uSy + 5|(y + u). Determine whether S is symmetric, reflexive…

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Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: Let R be a relation on the set of all functions from Z to Z defined by: R = {(f,9)|f(1) = g(3) or…

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Q: Let R be the relation on Z given by R= {(a,b) : a < b}. %3D (a) Find R-1. (b) Find R which is Z x Z…

A: relation on Z.

Q: 4. Define the operators T and U on the space P(R) of all polynomials in variable x as follows:…

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Q: Let S be the set of all real numbers and let R be a relation in S, defined by R = {(a, b) : a < b}.…

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Q: Consider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?

A: No it is not a reflexive relation

Q: Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3,…

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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: Define the relation A=((x,x+ 1):xER] Show that A, is a function from R to R.

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Q: Which of the following relations are functions of y in terms of x? 1) 2) 3) 4) 5- 5+ 5+ 4- 3- 2- 2-…

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Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

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Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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Q: b) Prove or disprove: i. [x]-[x]. ii. For all positive real numbers x and y, [xy] ≤ [x] - [y].

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