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: Method of algebra and cite a property used for every step: (Z − Y) − (Y − X) = Z − Y.

A: To prove: (Z − Y) − (Y − X) = Z − Y

Q: Prove: | a | - | b | <= | a + b |

A:

Q: Propositional Logic We say that two propositional formulas a and B are anti-logically equivalent if…

A:

Q: Let Ai = {i, i + 1, i + 2, . . .} find ∩∞(infitity)i=1Ai and ∪∞(infinity)=1Ai Explain.

A: We have to find the intersection and Union of given set.

Q: coprime

A:

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: determine which of the followingmatrices are nonsingular:

A: Given, 1322141-72 We say that a matrix is singular if its determinant is zero otherwise we say it is…

Q: Prove that for operators A and B which commute with their commutator [Â, Â]= Ĉ, so that [Â, Ĉ]=[ B,…

A: Given that A⏞,B⏞ commute with their commutator C⏞. Now, ekA.Be-kA=B+kA,B+k22!A,A,B+k33!A,A,A,B+...…

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…

A:

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: ∃x ∈ D such that ∀y ∈ E, x + y = 1 Negation:

A: We need to find the negation of  ∃x ∈ D such that ∀y ∈ E, x + y=1

Q: x + y)(z + w) = xz + xw + yz + yw A.) Explain in your own words what this problem is asking. B.)…

A:

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: FOR ANY TWOCOMPLEX NUMBERSZ, ANDZ,,PROVE THAT |Z|+|Z,|=,(Z, +Z,)+ /Z,Z» +÷(Z, +Z»)- Z,Z, .

A: for two complex number Z1 and Z2 prove that Z1+Z2=12Z1+Z2+Z1Z2+12Z1+Z2-Z1Z2

Q: snip

A: Inner product of two vectors is equal to sum of product of corresponding elements of the two…

Q: Theorem 2. Let a, b, c E Z such that gcd(a, c) = gcd(b, c) = 1. Then gcd(ab, c) = 1. %3| Proof.…

A: the gcd of two integers is the largest positive integer that divides each of the integers.

Q: SUMA Y RESTA DE NÚMEROS COMPLEJOS   Realizar las siguientes operaciones:          a).- Dado Z = 7…

A: The problem has been translated and solved. The translated question is  ADDITION AND SUBTRACTION OF…

Q: Let X= a, b, c, d, and classify each of the following as -algebra or not justifying all your…

A: Here X = {a, b, c, d}    Any collection à of subsets of X is a σ-algebra if it satisfies following…

Q: • Assume all sets are subsets of a universal set U. Suppose that A, B, and C are arbitrary sets,…

A: For problem a i have proved the both seth inequality. And for second i have used algebraic method

Q: DISPROVING. Let P,Q, R and S be propositions. Disprove: (PvR) ^ (P→Q) ^ (R→S)) = (Q v S)

A:

Q: Prove that |x|−|y|≤|x − y|. Hint: Apply the triangle inequalityto y and x − y.

A:

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

A:

Q: 6. show that the set A = {P= (₁4₁2) (R³ | 1²4 + √²2+x2 is an open out.

A:

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

A: