(1) Is p→ (q V r) logically equivalent to → (p Vr)? Prove your answer.

Q: Premises: Vx(L(x) → H(x)) 3x (~ J(x) A L(x)) Conclusion: 3x (H(x) A - J(x))

A: Given that Premises:      ∀xLx→Hx                        ∃x~Jx A LxConclusion:  ∃xHx A ~Jx We have…

Q: The statement [P (Qv R)] [(Q→ R) v (R P)] is

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Q: Prove that (C, +) ≈ (R × R, +).

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Q: If  A  is singular then det(A)=0 true or false

A: If  A  is singular then det(A)=0. We have to state true or false for this statement.

Q: Q #2 Let X - R" and ne N and z = 3 X,.G V neN where Gn = { (miy) €R* : y>nuf. gs z a topoligy on X.…

A: Topology means it should satisfy 3 conditions  Empty set and whole set should belong to tau.…

Q: |u| = |v| show that (u + v) • (v - u) = 0 is logically equivalent to condition u⚫u=v•v

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Q: EX+ Prove that [X Ju cx)] = x'Jr-, cx) Solp 2. ヘャut 2m+2V (-1)" X 2m+V (m!)P(m+v+1). neno 2m+2U-1…

A: Property of gamma function: Γ(m+1) = mΓ(m) ............. (1)

Q: - T:(x, y) → (x, y – 2). What is the preimage of (3, 5)? b. (3, 7) a. (5, 7) с. (3, 3) d. (5, 3)

A: Given transformation is T:x,y→x,y-2

Q: Consider an MA(2) Y; = po + 01ɛ+–1+02€t-2+ Et What are the values of the following quantities? (b)…

A: The most common type of stationary time series models is called as moving averages. yt =…

Q: t Evaluate [²(te² ² + ² + ₁ ² + √²=TE) 1 3 Ik dt. i

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Q: dt 20.5 t2+2t A - In|t? + 2t| + C B. 2ln +C C. In +C D. In + C

A: First we common out t2 from denominator and then put 1+2/t=u and differentiate both sides with…

Q: (b) y, (p → v) E » (Modus Ponens (MP)). Then apply MP to derive the converse of the Deduction…

A: Modus ponens, also known as modus ponendo ponens or implication elimination or affirming the…

Q: F.T. 3) If X(t) + y(t) = x (2t - 1) =it W+1 Ans: Yow) = [4+jCW+1)]² X(w) and X (w.) = -j(w+1) e 2…

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Q: 21 -2 T 5. If B= 1 -2 T = - +4 = 2 +5 1 Using two methods to find [T]B-

A: Method I- Let T be linear transformation on  a Vector space V with n dimension with basis…

Q: a(k) = eak coshw.k coshw, k ewok te-wok 2 eak coshw.k eak ewokteake-wok 2 X(Z) 3D글 z(enk e.k) + 글…

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Q: b) (i) Show that dz is a constant independent of z, and R. { Hint : z= Z, + Re"} 1 +- 2z-4 2 -z 3…

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Q: Prove that I(r + 1) = ar(r). 3. ii. Prove th at 3(m, n) = B(m + 1, n) + B(m, n+ 1). %3D

A: The required proofs are detailed below.

Q: S 1.Prove that L (coswt): %| s2 +w2

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Q: [2 11 Q2) Assume A = |1 [3 4] 2 4 11 3 6 4 3 and B = show that (AB)' B' A'

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Q: (a) Prove that de 2n a>b>0 Jo a+ bcose Ja? - b2

A: We will use basic knowledge of Integral Calculus and trigonometric identities to solve this…

Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: Let n be a positive integer and let u(x, y) = (x² + y²)". Show that (2²2²2 + 3²²) ₁ dx² дуг u(x, y)…

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Q: h2 |w(x) = Let h = x1 – xo, and w(x) = (x – x1)(x – xo). Proof that max %D 4

A: Let w(x)=x-x1x-x0 where h=x1-x0Prove that wx=h24.

Q: 2 16 v2 6. Prove that x V(2 – x) dr = 15

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Q: [x +1 0 2 1 2 (3)if A = 3 1 and B = 3 y -5 4 -1 -5 4 z - 4. Find x,у аnd z.

A: If two matrix A and B are equal then Each matrix of A and B has the same number of rows. Each matrix…

Q: 5. (8 points) Let A € M₂ (R) be given. (a) If det (A) = 1, prove that adj (adj(A)) = A. (b) If A is…

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

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Q: sv(8vh) (8→i)^(h→j) B. (i→j)→(fvh) .h Proof: Step Statements Reasons sv(gvh) (g i)^(h→j) gvh iv j…

A: We are given the following Logic: f∨(g∨h)             : Premise 1(g→i)∧(h→j)    : Premise…

Q: Prove that for any x > 0, y>0 (x² + y²) ³ ≥ (x³ + y²³) ³. Hint: Introduce a new variable z = (2)².

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Q: (3x)[R(x) V S(x)]→(3x)R(x) V (3x)S(x)

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Q: Let f(x)= x+1, find all xeR such that f[f(x)]=2 the solution set is: а. у%3D0. b. y=x. с. у#х. d.…

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Q: Reason ер Vx(P(x)v Q(x)) P(a)vQ(a) P(a)vQ(a)v R(a) Premise Vx(-P(x)^Q(x))→R(x)) (-P(a)^Q(a))→R(a)…

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Q: k – 1 k+1 Let h : Z → Q × Q, given by h(k) : 2 2 Let G = {(m, m +1) | m E Z}. Show that G C h(Z).

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Q: T(7)T(m) B(m,m) for m >0 22-T(m+) 21

A: we know that βm,m=ΓmΓmΓm+m=∫01um-11-um-1du ΓmΓmΓ2m=∫01um-11-um-1du Let u=1+x2 so, du=dx2 and…