Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

Q: Show that the function h: AUB → {1,..., m+n} in the proof of the first counting principle, defined…

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Q: (a) Prove that every element of Q/Z has finite order.

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Prove by structural induction that F C F2

A: Given that, F1 such that, 3∈F1, n∈F1⇒n2∈F1. and F2 such that, 3∈F2, n,m∈F2⇒nm∈F2. Consider Pn be…

Q: Prove B = {1/n : n in Natural numbers} is a union of {0} and that it is compact.

A: A set B is said to be compact if every open cover has a finite subcover of B.

Q: 1. (p.209, #16) Define a relation R on Z by declaring that xRy if and only if x? = y? (mod 4). Prove…

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Q: Prove that Z is countable, exhibiting an explicit bijection between N and Z.

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Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

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Q: 1. Prove that the followingn is a theorem in SD. Av - A

A: Theorem :-

Q: Prove that the class {AC – B, BC – A, AB − C} is pairwise disjoint.

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Q: Prove that Q is not isomorphic to Z.

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Q: Prove that z =z iff z is real.

A: z represents a complex number which is in the form x+iy, where x and y are the real numbers and i is…

Q: Prove that d(A) = Ø if and only if A is both open and closed.

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Q: Prove that S4 is not isomorphic to D12.

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Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: Prove that R under addition is not isomorphic to R* under multiplication.

A: we know that R and R* = R – {0} are groups under addition and multiplication respectively. We have…

Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

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Q: 1. Prove that if A is infinite and a E A, then A – {a} is also infinite.

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Q: Prove that the following is one- to one and onto:  f(x) =x3+2

A: To verify that the given function is one -one onto from R to R (R= the set of real numbers)

Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

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Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: Prove that R is not reflexive.

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Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

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Q: Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

A: Let, a and b belong to Sn.   Consider α is the product of a 2-cycles, and β is the product of b…

Q: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

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Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

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Q: Prove that if H ≤ G, then aHa – 1 ≤ G, for all a in G.

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Q: Let α and β are elements of Sn. Prove that αβ is even if and only if α and β are both even or both…

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Q: Show that for every x E R, there exists an n EN such that <x<n.

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Q: 7. Let E and 0 the set of even and odd integers, respectively. Prove that {E, O} partitions Z.

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Q: Show that given relation is transitive For X, y € R say x is congruent to y modulo Z if x – y is an…

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Q: For any set A and any ≥0, there exists an open set O such that A CO and m* (0) ≤m* (A) + €.

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Q: Let A E Mnxn (F) and B = aA, with a E F. Prove by induction that det(B) all positive integer n. a"…

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Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: To be onto, for any integer y, there exists an integer x such that y = f(x). Give a counterexample,…

A: To be onto for any real number y, there exists a real number x such that y = f(x). Let y be a real…

Q: Prove that if gcd(a, b) = 1 and cla, then gcd(b, c) = 1.

A: We need to prove If gcd(a,b)=1, then gcd(b,c)=1. The proof is given below.

Q: Prove the follawing identities. 4-V. 4- V = AV (a) (b) M8 = A/1+ 4) (1+ A)" 2. (C) 4/f = I for Afn-…

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Q: Prove that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: Suppose that E is an extension of F, and a,b are elements of E. If a is algebraic over F of degree…

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Q: Prove that if a is an upper bound for A, and if a is also anelement of A, then it must be that a =…

A: We have to prove that if a is an upper bound for A, and if a is also an element of A, then it must…

Q: Prove that X ∈ ℘(A) if and only if X ⊆ A

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Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Define H : Z → Z by the rule H(x) = 4x – 5, for all integers x. (Z: set of integers) (a) Is H…

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Q: 5. Let X, be a RSR (mod n). Prove that E e(E/n) EXn is a multiplicative function of n, where e(x) =…

A: Given Xn be a RSRmod n. We know that RSRmod n=x∈ℤ: 0&lt;x&lt;n, gcdx,n=1. Now consider the function…

Q: Let A E Mnxn(F) and B = aA, with a E F. Prove by induction that det(B) = a" det(A), for all positive…

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Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

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Q: Define a set X of numbers as follows. В. 2 E X. R2. If x E X, so is x+ 10. Use induction to prove…

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Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

A: (a) Given that  f:X→Yg:Y→Z and g∘f is onto To prove : g is onto Let x∈X, y∈Y , z∈Z As gof is onto…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.