4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b) Prove that mn | a. Let a, m, n E Z. Suppose that m and n divide a and that (m, n) = 1.

Q: U is the set of the numbers on a 6-sided die and B is the set of numbers on a 6-sided die that are…

A: Given If U is the set of the numbers on a 6-sided die and B is the set of numbers on a 6-sided die…

Q: Let ū = (-1,2,4) and ū = (-3,5, 2). Compute the following: (a) ū+ū (b) .공 (c) ||해 (d) ũ - 7 (this is…

A: (a) Find u→+v→ as follows: u→+v→=-1,2,4+-3,5,2          =-1+-3,2+5,4+2          =-4,7,6

Q: Let A=(1, 0), B=(0, 1), C=(1, 1), and D=(r, 1). Verify that R(A, B, C, D) = r.

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Q: Let a, b c and d be real numbers. Show that ab-cd = b(a-c)+c(b-d), and deduce that |ab - cd| ≤…

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Q: show that }I f,lm) Ck-m) 2.

A: For, ∑m=0kf1mf2k-m, substitute k-m=n or m=k-n: ∑m=0kf1mf2k-m=∑k-n=0kf1k-nf2k-k-n…

Q: 4. Prove for A = {x: x = 12k, k = Z}, B = {x : x = 3k, k = Z}, and C = {x: x = 4k, k = Z} that A…

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Q: 25. Show that |v|u + Ju|v and |v|u – |u| v are orthogonal.

A: Two vectors are orthogonal if their dot product is zero. If the dot product of two vectors is zero,…

Q: 6. Prove that if alb and k E Z, then a|(b+ ak).

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Q: 1. Prove the cancellation property of addition (If x+ z=y+z, then x =y). 2. Prove: += ad/bc. 3.…

A: "Since you have posted a question with multiple subparts, we will solve the first three subparts for…

Q: Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

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Q: P(X=x) 0,10 0,20 0,30 0,40 P(3 < X S4 )= ??? O A) 0,10 O B) 0,40 C) 0,30 O D) 0,50 XN34 5

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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: (а b)с + (а - с)b + (с b)a 3 ve that v belongs to B.

A:

Q: Let a, b, q, r E Z and assume that a and b are not both zero and b and r are not both zero. Show…

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Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Show that a,b,c are in A.P. If x is the G M be tween a and b and y is the G M between b and c,  then…

A: We have to show if  a, b, c are in A.P.,  x is the G M be tween a and b and y is the G M between b…

Q: Let r ERQ. Prove that for all a, b, c EQ, =cr iff a = 0 and b = c. a + br

A: Given that r∈RQ Let a,b,c∈Qa+br=cr iff a=0 and b=c

Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

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Q: Let a, b, c € Z with c‡0. Prove that if a | b, then ac | be

A: Let a,b,c∈Z and c≠0. Let ab. Then b=ak, for some integer k.

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: 2. Let m, n, a, b € Z. Prove that if 6 | m and 6 | n then 3 | (am + bn +15).

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Q: all your calculations and simplify your answer completely. Il you only give a final an- Swer you…

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Q: If P(A/B) = 0, then A &B are: Select one: a. Mutually exclusive b. ACB с. ВС А d. Independent

A: There are two events A and B such that PA|B=0 PA|B=PA∩BPB=0 Therefore it follows that PA∩B=0

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Let _X=R². Find M² if M= {(a,1)}.

A: Given that, Let X=R2 Find M⊥ if M=a,1.

Q: 9. Let a, b, and d be positive integers. Prove that if (a, b) = 1 and d divides a, then (d, b) = 1.

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Q: a. Let U = { a, b, c, d, e, f, g} X= {a, c, e, g} , Y = { a, b, c }, and Z = {b, c , d, e, f} a1.…

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Q: 1. Let a, b, c be positive real numbers, and let p, q, r be [0,1/2]. Suppose a+b+c= p + q + r = 1.…

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Q: 2 1] Assume A = 1 3 4. [2 2 4 and B = show that (AB)' = B' A' [3 6 4]

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Q: If a € R such that 0 0, then show that a = 0.

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

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Q: Suppose u, v belong to V. Prove that = 0 if and only if ||u|| ≤ ||u+av|| for all a belong to F

A: We have to prove  if and only if

Q: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: ). Find 1 |z-j|=2 Z

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

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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: 15. Suppose that a, b e Z and gcd(a, b) = 1. Prove that ged(a", b") = 1 for m, ne N. N and lot 8 frm…

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Q: Let A={2,y,z} and B={3,x,z}. How many distinct elements are in (A×B)∩(B×A)

A: Given, A={2,y,z} B={3,x,z} How many distinct elements are in (A×B)∩(B×A) ?

Q: 26. Let A=61] Find a) A V B. and b) Α Λ Β. B = [id] c) A OB.

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Q: Let a, b, c E Z. Prove that if (a, b)= 1 and cl(a² + b), then (a, c) = 1.

A: Given gcd(a,b)=1  So 1 can be expressed as linear combination of a and b  That is 1=ax+by…

Q: Let a, b e Z so that 2a + 36 is a multiple of 11. Prove that 11 divides a - 4b.

A: