1. Let A = {a,b,c} and B = {1,2, 3}.a) Write out all functions f : A → B using two-line notation. How many differentfunctions are there, and why does this number make sense? (You might want to considerthe multiplicative principle here).b)  From part a, identify the injective functions. Identify the surjective functions.

Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

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Q: Let In, f be real fun ction on 52.Show {w : fu(w) yo frw) }=Ŭ o U {w-1fu(w) %3D f(w)1>と3

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Q: 6. Show that if f : X Y and g : Y → Z are both onto then fg : Z -→ Z is onto. (Note this is function…

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Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

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Q: Show that R2 =

A: Then, V =Span(S), if every vector in V can be written as a linear combination of elements of S.

Q: 3. Find and sketch the domain of f(x, y) = In(9 – x² – 9y²). [Hint: In(x), x > 0]

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Q: f) Is the relation R, defined on the set of real numbers by xRy if 2x + y = 3. antisymmetric? g) Is…

A: Use the definition of antisymmetric to check whether the given relation is antisymmetric or not.…

Q: Prove that the set A = {x ∈ R | 3x + 2 ≤ 0} has a maximum

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Q: A relation R is defined on the set of real numbers by xRy if and only if x-y is a multiple of 3.…

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Q: let G={2nEIJ.] show that (I, +) = (G₁.)

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Q: Let x={1,2,3,4} and y={1,2,3,4,5,6,7,8,9,10}   and "f" from X to Y is given by…

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Q: 4. Let R be the relation in the positive integers

A: 4. Given, R be the relation in the  positive integers ℕ defined by                        R=x, y: x,…

Q: Assumed that F OG = Ø, show that we can express the infimum and supremum of F NG in terms of the…

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Q: 2. Let f : (0, –→R such that f(x) = x². Show that f(x) < r. • Show that f has no fixed point on (0,…

A: The given function is f:(0,13]→R and defined as f(x)=x2 . we have to show that f has no fixed point…

Q: Show that the function g: R -R defined by g(x) = 4x2 +3 is neither one-to-one nor onto

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Q: Compute f. F · dr, where F(x, y) = (8 + 21x², 4xy) and C is a line segment from (0,0) to (1,2). (Use…

A: Given F(x,y)=8+21x2,4xy2        (1) Line segment from (0,0) to (1,2) is y-0=2-01-0(x-0)⇒y=2x…

Q: Find and sketch the domain of z = f(x, y) = In(9 - x - y²).

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Q: Is the relation R, defined on the set of real numbers by xRy if 2x + y = 3. antisymmetric?

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Q: 1. The set of all pairs of real numbers of the form (x, 0) with the standard operations on R2.

A: 4. Given that the set of all pairs of real numbers of the form (x, 0) with the standard operation on…

Q: Let A = {3, 4, 5} and B = {1, 2, 3, 4} and define a relation R: A → B as follows: For all (x, y) ∈ A…

A: A=3,4,5B=1,2,3,4R=x,y: x∈A, y∈B ∋xy&gt;7 a. R=3,3, 3,4, 3,5, 4,2, 4,3, 4,5, 5,2, 5,3, 5,4, 5,5

Q: Let A = {3, 4, 5} and B = {1, 2, 3, 4} and define a relation R: A → B as follows: For all…

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Q: Let A and B be set .Show that AxB → BxA Such that (a,b)=(b,a) is bijective function

A: A function f from a non-empty set A to anther non-empty set B is said to be an one-to-one or…

Q: Let A = {x | 5x + 1 E Q} and B be the set of all even integers. Prove that A and B have the same…

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Q: et F be the collection of

A: To define, T(G)=∫G(z-y)dG(y),G∈F

Q: Prove that if f(z) is analytic, then: [f'(2)² = (² |f' (2) 1³² = (²+ (²)² = (²+)²

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Q: Prove that for functions f : R→R, the e- 6 definition of continuity implies the open set definition.

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Q: Compute f.F. dr, where F(x, y) = (2+ 12x²,9xy² ) and C is a line segment from (0, 0) to (1,2). (Use…

A: To Determine: Compute ∫cF.dr,where Fx,y=&lt;2+12x2,9xy2&gt; and C is a line segment from 0,0 to 1,2.…

Q: Prove that there existo a bijective function between the set of all integer numbers Z and the set of…

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Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

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Q: 7. Compute gfg, the conjugate pair of f by g for f= (2, 3, 5, 4) and g = (1, 3, 2)(4, 5).

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Q: 6. Based on your work above, and what you know about the multiplicative principle, how many…

A: The one-one function is the function each element of B  has a unique preimage of A. Also, know as…

Q: be an integral domain and define a relation ~ on xDsuch only if ad = bc: Prove that the relation is…

A: Solution

Q: 4. Let R be the relation xRy if 3| (x – y) on Z. Prove that R is an equivalence relation.

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Q: Show that if for a relation R we know that R2 C R, then R transitive, and conversely. Ent. There are…

A: To prove whether a relation is transitive or not

Q: 9.15. A relation R is defined on Z by a Rbif |a – b| < 2. Which of the properties reflexive,…

A: Given : A relation is R is defined on Z is given by  a R b if a-b≤2  Which of the properties…

Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: Sketch a possible function that satisfies the following conditoons: a) Quartic, as x goes to…

A: It is known that a polynomial function with n degrees has at most n zeros and n-1 turning points.…

Q: prove R is not homomorphic to R^2

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Q: Obtain the 16 relations of the set {0,1}, indicate which of these are reflexive, symmetric and / or…

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Q: 5. The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard…

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Q: It was noted that the binary propositional operators, A and V, can be regarded as functions, f: B x…

A: For any binary operator, number of rows in the truth table =4 Each row can have either True or False…

Q: (a) How many functions are there from {1,2,3,4,5} to {A, B, C}? (b) How many onto functions are…

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Q: Given any set of 15 integers, must there be two that have the same remainder when divided by 12?…

A: We have 12 remainders when a integer is divided by 12 those are 0,1,2,3,4,5,6,7,8,9,10,11   Let us…

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: Do you agree with the contention that the funtions f(x) = x + 2 and g(x) = x2 -  4 ÷ x - 2 are the…

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Q: Take the following derivation rule in predicate calculus: Vr.P(x)Q(x), Vr.Q(x) → R(x) Vx.P(x) → R(x)…

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Q: i Determine whether or not the relation R on the set of all real numbers is reflexiv netric,…

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Q: 2. A relation p is defined on the set of all real numbers R by xpy if and only if |x- yl < y, for x,…

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Q: recall the definition of |A|=|B| for sets A and B and use that to show  |N|=|{ n/2 : n∈ Z}|  no…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.