Theorem 20 (Associative Law). If x, y, z ∈ N, then (x+y)+z = x+(y+z). Proof. (sketch). This follows from Lemma 17, and the identity A ∪ (B ∪ C) = (A ∪ B) ∪ C. Exercise 9. Write up the above proof. (You do not need to prove the identity A ∪ (B ∪ C) = (A ∪ B) ∪ C, since it is part of basic set theory.)

Q: Show that -p → (q →r) and q → (p V r) are logically equivalent. Show your work and justify each one…

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Q: Prove that (ab + cd)² ≤ (a² + c²)(b² + ď²) for all a, b, c, d ɛ R.

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Q: Prove the following set identities: (a) (AUB) nC = (ANC) U (BNC); (b) (AUB) - C= (A-C) U (B-C); (c)…

A: The prove of the identities (a), (b) and (c) are given below.

Q: 3 Let x = (9, −9, −4) and y = (-6, 6, -9) be in R³ and a be in R. Then ax + y = ( ,23).

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Q: Prove that for any sets A, B, X - X − (An B) = (X − A) U (X – B). -

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Q: Which of the following is always true about the Cartesian product of sets A and B? A. (A x B) x C =…

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Q: 1. Let X be the set of real numbers with the usual pseudometric. Find the derived set and the…

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Q: Suppose x and y are integers. Show that If x y, then x | 3y - y + 5y. 1) What is your assumption? 2)…

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Q: Find a pair of values (x, y) that serve as a counter-example for the statement that ⌈xy⌉ = ⌈x⌉⌈y⌉…

A: Let, x = 2.1 & y = 2.5

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: (a+b)(a+ba)(ao³ + bw) = a' +b*

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Q: III. Determine whether the statement is true or false. (Example: if B+2 10, then 4 37 is FALSE…

A: Given that:                            1. -15<152. 8+3=11 Or, 5 is an even number3. 1=2 and 15 is…

Q: Use the relationship you described (or the one given in the solutions to Homework 2) to prove…

A: Let the two numbers be 'a' and 'b' Let the GCD of the numbers 'a' and 'b' be 'G' Then we can write…

Q: If x1 = x and x2 = x + Ax, what is Ax?

A: We find ∆x as, x1=xx2=x+∆x⇒x2=x1+∆x⇒x2-x1=x1+∆x-x1⇒x2-x1=∆x⇒∆x=x2-x1

Q: Use any method you choose to shc are sets.

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Q: Let x) 4 and gx)-x-3. Find the foilowing lock

A: To find the following expressions

Q: 4. Show that Archimedean Principle is a consequence of the assertions that every interval (a, b)…

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Q: j+ky-o Find k Iet yco)=lo0 andyc2)=4 3D0

A: Given that y'+ky=0 y(0)=100 and y(2)=4

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…

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Q: Find a pair of values (x, y) that serve as a counter-example for the statement that [ry] : |x||y for…

A: The ceiling function gives smallest integer that is greater than the number, that is, x is the…

Q: 1. (a) Prove that if alb and alc. then alb² + 10c + a.

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Q: Proof by contrapositive that for x, y ∈ Z, if xy is even then either x or y is even.

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Q: prove the following statement.

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Q: (A x B) U (C x D) (AUC) x (BU D)

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Q: Let A, B, and C are three sets, prove that Bx(AUC)=(B× A)U(B×C)

A: "Since you have posted multiple questions, we will answer the first question. If you want the…

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: Let A {a, b} and B = {1, 2}. Give an element of A x B x A.

A: Let M and N be two sets. The Cartesian product, M×N is the set of all possible ordered pairs with…

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Q: Proof the following statement in detail: ". Let x ∈ R. Prove that if 3x4 + 1 ≤ x7 + x3, then x >…

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Q: 1- Prove that (a) |A× B|² = |A|2|B|? – (A · B)² (b) (Ax B)(C × D) = (A· C)(B· D) – (A · D)(B · C)…

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Q: Prove the following set identities: (a) (AUB) nC = (ANC) U (BNC); (b) (AUB)-C=(A-C) U (B-C); (c)…

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Q: Which of the following statements are true? 1. sec 2x = sec 2 sec x II. sec 2æ = sec(x +æ) III. 2…

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Q: (b) If a > b and c > 0, then ac > bc.

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