c) Find inverse of f(x) 0. Define S={x|x mod 4 =3} and T={y| y mod 2-1}. Prove, Sis a subset of T

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

A:

Q: Verify that for all x, the change-of-base formula holds: bx = ax Ioga b for a, b > 0 such that a…

A:

Q: 13. Prove by Math Induction ex > 1+x for all positive integers x > 1

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

A:

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: 15) Find The natural domain (a) fx) =3 os X 6 g0) = ©dix) = Sin? x O ra) = 19-x2.

A: given(a)f(x)=3cosxdomain of this function is (-∞,∞)(b)g(x)=13+xx≥0therefore domain is [0,∞)

Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

A:

Q: 1. Let X = {z €R:r#0,1} and define f : X X by f(r) = 1-. Prove or disprove that f is a permutation…

A: As per our guidelines we are supposed to answer only first one. kindly reposed other part in the…

Q: 2. Let A € M₂ (R) be given. (a) If AA¹ = 0, prove that A = On. Does this hold if A € M₁ (C)? (b)…

A: This is a problem of Linear Algebra.

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

A:

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

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Q: The characteristic of Z5[x]/ is O 3 5

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Q: . Prove the following: If gcd(a, b) = 1 and ca then gcd(b, c) = 1

A: We will prove the given statement.

Q: Prove the following identity with a combinatorial proof: nz m) n+ K t 1 こ i=0

A:

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

A:

Q: Prove that if A, B ϵ Mnxn(F) are similar, then det{A) = det(B).

A: Given:  A, B ∈ MnxnF are similar To prove:  det(A)=det(B).

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

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: Prove that if n> m - 1, then J,(x) = 2(x/2)"-m

A:

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: (12) f(x) has inverse iff f(x) O onto O one to one O one to one and onto

A: A function f is invertible if and only if it is both one to one and onto.

Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: (b) Use part (a) to show that f (T) is either a transposition disjoint transpositions. or a product…

A:

Q: Prove that if Tr(AT A) = 0, then A = O .

A: Proof:- Suppose A is a square matrix of order n, That is, A=aij For 1≤i,j≤n Now we need to prove…

Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: 1. Let f:Z - Z, defined by f(2) = 2 (mod n), V z E Z, then kerf is (B) nz (A) (0) (C) Z (D) (0,1)

A: We are supposed to answer only one question.

Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

A:

Q: Prove by Contradiction that if    a 0, which implies  a< = b

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Q: If gcd(a, c) = 1 and gcd(b, c) = 1, prove that gcd(ab, c) = 1.

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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-…

A:

Q: 1. Let A = {x E Z | 6 divides x} and B = {x € Z | 3 divides x}. Prove that A C B, and then prove…

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Q: b) Prove, by induction, that for any k > 1 and any choice of c1, ... , Ck E R and X1, ... , Xk E V,…

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Q: v. Let Ac S, prove that P(Ā) = 1- P(A)

A: Given that

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: The expansion of esin(x) is Select one: O a. x² x4 1+x+ ... 8 Ob. x² 1+x+- 2 - t... 8 c. x5 1+x+ 6.…

A: To find the expansion of

Q: 3. а) Prove by induction that 1 P(n):2+6+12+20+...+n(2n+2) =n(n+1)(n+ 2) Vn21 1 3 Let f:R→(-1,1) be…

A: For part a, principle of mathematical induction is used. For part b, function is defined when imagee…

Q: 6. If X has full rank, so that P = X(X'X)-1X', prove that C(P) = C(X).

A: The rank of a matrix denotes the number of independent rows or the number of independent columns in…

Q: 3. а) Prove by induction that 1 P(n):2+6+12+20+...+n(2n+2) = n(n+1)(n+2) Vn21 2 1 3 Let f:R→(-1,1)…

A: For a, principle of mathematical induction is used. For b, function is not well defined as images of…

Q: When A has full rank, the pseudo-inver en m≥n, and as A+ = AT (AAT)-¹ when n A = I, so A+ is a left…

A:

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 * b = 2ab be defined on Z*. 1. Is * Commutative? 2. Is * Associative? 3. What is the identity…

A:

Q: ,,E R, nxn mathix, Such that For nZ 2 and any X,,X2, if P dis an nxn mattx such thet P dis (P);;…

A: We have to prove the given statement by induction.

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

A:

Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

A: In determinant the scalar quantity is taken common from a single row or a column.

Q: Prove that if A,B ∈Mn×n(F) are similar, then det(A) = det(B).

A: If the matrices A and B are similar then there exists a non singular matrix S,

Q: Q3. Prove that if kx = ky (mod m) , then x = y (mod m/d) where d = gcd( m, k).

A:

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below