Let a and ß belong to S,. Prove that aß is even if and only if a and B are both even or both odd.

Q: b let fl) = an+b and g) = Cn+d where a,b;c and d Constant. Detemine for which constant a,b;c and d…

A: The objective is to find the constants a, b, c and d such that fog=gof where fx=ax+b and gx=cx+d

Q: If A and B can be placed in a one-to-one correspondence, must A=B?

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Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

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Q: If A = (; A-: )-..R. e M2x 2(R), find A" for any positive integer n.

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Q: Let 4; = {0, i} . Find U4, and N 4. i= 1 | A;. i= 1

A: Given :  Ai =  0 , i    To find :  The union ∪i=1∞ Ai    and the intersection  ∩i=1∞ Ai   .

Q: Are u sure its not 1

A: It is given that a=0 and b=1.

Q: 2.If alb and alc then al Cbx tcy) for all X>YEZ.

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Q: 5. If x is an integer, then x' 2x. b d a C 6. There exists a real number x such that x'= x . FL

A: we can solve this problem using some basic properties of integers and real number.

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: A. j2n = (-1)" B. jan+1 = j C. j¯" = (-j)" D. jan-1 = –1

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Q: Can you find a,b,c, and d (if exists) such that AB = BA.

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Q: If (R – {0}, .) = (Q,.) then 1=0 1=-1 -1=0

A: For the given statement

Q: Ema thinks that (u) xw, exists. Paul disagrees with her. Who is right? Explain why.

A: Dot product gives a scalar value  .

Q: Show that if AB = I, then det(A) 0 and det(B) # 0.

A: Here given as AB=In find determinant both sides detAB=detIndetAB=1

Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

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Q: If z1:2., are complex numbers such that =|= =. = |=,| =1, Prove that

A: To prove: z1+z2+…+zn=1z1+1z2+…+1zn Given: z1=z2=…=zn=1 Since, (1) zz¯=|z|2 (2) z=z¯

Q: Suppose that p is false, but q is true. Which of the following statements are also true? Select all…

A: Given the statement p is false, but q is true.

Q: If a1, az, az,.-ans. be the A.P. and S, = a-1m+1+a-1m+2+a-1n+3 +.--+ am %3D Prove that :…

A: In this question we have to proof S1, S2, S3,.... are in Arithmetic Progression.

Q: 6. If -2 determine all values of x and y for which A= A.

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Q: 10.

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Q: Let à . Find al.

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Q: Let ---0--0 8 u= W= Compute 3w+6u = -131

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Q: Suppose gcd(a, b) = 1. Prove that ab | c if, and only if a|c and b| c.

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Q: Suppose that P(A) = 0.4, P(B) = 0.3 and PAUB)-042, then a. A and B are dependent b. none of the…

A: We have to find out correct answer...

Q: Suppose gcd(ab,c)=1. Prove that abc | d if, and only if ab | d and c | d.

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Q: k of [Adj A], it -i 0 i

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Q: (1) if is reflexive, symmetric, or transitive. For a, b = Z, define ab if a + b is a multiple of 3.…

A: A relation is a subset of the cartesian product A×B where A and B are non-empty sets. For cartesian…

Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

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Q: - [a C b If A = where a, b, c and d are real numbers, then prove that A²-(a + d)A + (ad-bc) I=O.…

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Q: l, tet that 17/2n+3m e integed, prouve iff, 17/9n +5 m and m be

A: Properties used: Property 1. If x divides y, then x divides ky and vice versa. x|y⇔x|ky Property 2.…

Q: Let ü = 21 – k and i = i + 3j + k. Compute |ū], |ū], ū × ĩ, and Projzu.

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Q: Suppose a > 0, (a, 42) = 6 and [a, 42] = 420. Find а.

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