Describe a bijection in each of the following cases. In each case it is enough to describe a suitable bijection. You don't need to justify that it actually is a bijection. (a) (b) (c) (d) From w= {0, 1, 2, ...} to N = {1, 2, 3, ...}. From Z to N. From the interval [0, 1] to the interval [—1, 1]. From the interval [0, 1) to the interval (0, 1]. From R to R\{0}.

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: prove if x → y then xor y = y =

A: To prove : If x→y then x or y = y Lets use the help of truth table to prove the above statement.

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

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Q: EXvyP(x, y) is true when there is an x for which P(x, y) is true for every y. A True B False

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Q: Evaluate (Vx+ f. d dx. OA. A. Vx2 + 2 + a² - a In + C O B. Vx2 + a² - In |x| C. Vx2 • a? + In- + C…

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Q: Determine the interval(s) on which x2 + 2x – 3 > 0. - Ох1 Ox 1 -3 1

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Q: Prove |°z 'z|>|°z|-|'z|

A: We have to prove  z1-z2≤z1-z2            orz1-z2≥z1-z2 NOTE : The polar form of a complex number…

Q: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

A: Bijective function

Q: 2) Let R' real numbers, define = x,Y1 + 3x2y; + 5x3y3 for x = (x, X2, X3) and y (y,Yz, Y3). Is…

A: Given x=x1,x2,x3, y=y1,y2,y3 in ℝ3. Define x,y=x1y1+3x2y2+5x3y3

Q: Suppose that the translation T is described by (?,?)→(?+2,?−1)(x,y)→(x+2,y−1). What translation…

A: We have to find the translation described in x,y→x+2,y-1.

Q: Let f AB, prove the following: (a) if S, T are subsets of A, then f(SNT) ≤ f(S) nf(T); (b) if U, V…

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Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B)n (AUC)

A: To prove:- Prove that for any sets A, B, C we have AU (B∩C) = (AUB)  ∩(AUC) Method :- I use venn…

Q: Let {(x, y) E R² : |x| 1}. {(x, y) E R² : 0 < x < y}, {(x, y) E R² : 0 < y <¤ < 1}, U {(7, y) E R² :…

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Q: Find the of im < Re z< £ un der w-

A: Substitute w=u+iv and z=x+iy in the transformation. w=1zu+iv=1x+iyu+iv=x-iyx+iy.1x-iyu+iv=x-iyx2+y2…

Q: (a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational…

A: Given the notation , and using this defining the set . Then the function  defined as:.It is well…

Q: Given R = a) b) 24 15 and S = 2STR 5R+S-(2STR)T 1 3 15 Find

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Q: If X, Y, Z ER and x≤Z, show that XL Y S z if and only if |x-y| + | y-z| = |x-z). Interpret…

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Q: 4cos20 + 2/3 = 0 on the interval[0, 3t]

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Q: If f(x, y) = 2√/16 - x² - y², then the domain of f is the set D = {(x,y)|x² + y² ≤ 16}. Select one:…

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Q: If |z-8 <, and ly-히 < , then a - y < 1. 2'

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Q: 2. Let f(x, y) = (2x – y,x – 2y); (x, y) E R × R, (R is set of real numbers.) a) Show that fis one…

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Q: Solve the follawing yetem using augmented ito Matrisc M! C + + 32=1 *+34-2= 3 + Z=7 2x+

A: The given system of equations is x+y+3z=12x+3y-z=35x+7y+z=7 Solving the given system by converting…

Q: 2b. In R show that : || x - y || || x + y || ≤ 11x11 ² + 11y11²2² -(1

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Q: On which interval ƒ(x) = − 3x is increasing? 4 O a. (-∞, -2)and(−2, 2) O b. (-2,2) c. (-∞, -2)and(2,…

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Q: (b) Consider the statement: For all x, y, z ∈ Z. At least one of x− y, x− z and y − z is even. (i)…

A: Steps in roadmap for proof by contradiction for the given statement are: Step1: Assume that…

Q: prove that two vectur ae orthognal A = x + 4y+32 B =4x+2y-42

A: See the attachment

Q: Find: ii) P (X₂ ≤ 1X₂ ≤). f(x₁,x₂) = {1x1x² 0≤x≤1, 0≤ x₂ ≤ 1 elsewhere

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

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Q: (h) Vx E R, 3y E R, (x + y > 0 or x + y = 0). (i) Vx E R, 3y E R, (x + y > 0 and x + y = 0). (j) (Væ…

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Q: Transiate the graphical representation t0 an incquality using the vaviable x. 10 12 14 4- + 30 40 45…

A: Inequality for the first graphical representation is  x1 =14

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: B = {1 + x + x², x + x², x²}. for P2. Let p(x) = 1 + 3x + 2x². Then [p(x)]B = 2 of 2 1 None 1 2

A: The given set is: B= 1+x+x2, x+x2, x2

Q: Prove that |Re(2−7+z²)| ≤ 4 when |z|≤1.

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Q: Q1. Let f,g : R → R be given by f(x) = 6x – 10 and g(x) = 4x+5. Find the domain of f(x) h(x) =…

A: Given             f(x)=6x-10           g(x)=4x+5            h(x)=f(x)fog(x)…