Prove that if C is a point of −→AB other than A, then −→AC = −→AB.

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Show that Q(i) is a subfield of c

A: Subfield of a field

Q: Need prove If R is a UFD and f(x) , g(x) are elements of R[x] , then c(fg) = c(f)c(g)

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Q: Prove that K = {x+yv3 ] x, y E Z} is a subring of R.

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Q: Let B= {x €R|x E Q} be a set. Is B countable? Justify the answer by showing the explicit formula for…

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Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: The zero-product principle states that if AB = 0, then________.

A: Zero-product principal:

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

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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 using the IVT. √c +√c + 2 = 3 has a solution.

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Q: Prove that f : R – {3} → R – {1}, defined by f(x) = is a bijection. x-1 x-3

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Q: (5). If A, B and A+B are each non-singular and the products are conformable, prove that (a). A(A +…

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Q: Determine the value of c so that the pmf of X is valid. 2 cx :-1, 1 P(X) = { c (x + 1), x = 0,2

A: Answer - Given PMF of X                 P(X)  = cx2x= -1, 1c(x +1)x = 0, 2 Find the value of c

Q: Let A, B, C, and D be sets.

A: First, (AxC)U(BXD) ⸦(AUC) x(BUD)Take left hand side first , let  (x,y) ∈(AxC) U(BxD)=> (x,y)…

Q: Prove that R is not reflexive.

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Q: T be the lineas translfomation et whose standasd matsine is A= 2 -2 2. - 2 Does T maps 3 3. R4 onto…

A: A=2-29203-230005

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: U prove that In (l+x)) 2.

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Q: verify that (AT )T = A

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Q: Prove that, IXI<a Rf and If -aL X Z 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: Prove that X and Y are independent if and only if R = 0

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Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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Q: Prove that if x+y=%30 then y=-x

A: See the attachment

Q: The zero-product principle states that if AB 0, then

A: If AB = 0Then zero-product principle states that this is possible only if - A = 0andB = 0

Q: Prove that if A ϵ Mnxn(F) has two identical columns, then det(A) = 0.

A: Given: A∈Mn×nF i.e. A is a matrix of order n×n.   Consider a matrix with first two column as…

Q: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

A: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Let T be the linear thansformation from R5 to Rª defined by X1 1] X2 X1 1 3 2 X2 -8 1 -2 2 0 -1 4 X3…

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Q: Show that ∗ : R × R → R defined by        a ∗ b = a + 2b is not commutative.

A: Show that ∗ : R × R → R defined by        a ∗ b = a + 2b is not commutative.

Q: Find a zero-divisor in Z,[i] = {a + bi | a, b E Z3}.

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Q: Let x, y ∈ Z. Prove that x − y is even if and only x and y are of the same parity.

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Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

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Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

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Q: Prove that for the sets. B={4x+3|X€Z3 we have A =B

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Q: Prove that ||ư + B|| = ||T|| + ||B|| if and only if i = kủ. %3D

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Q: Let R→R be defined by g(x) 2y+1 2y+2 Is g bijective? Prove or give a counter example.

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: Use Green Theorem to Evaluate y x²dx – x²dy where C is left half of x² + y² = 25

A: Note that left half of the circle x2+y2=25 means the second and third quadrants. Now, in the second…

Q: Prove that R (D-1 and R)%=(-)

A: The objective is to prove that  Pn1=1Pn-1=-1n

Q: Determine whether T={(a, -a) | a EZ} is a subring of ZxZ-

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Q: Let S = {(x, y) :x² + (y – 1)² < 1} U {(3,0)}. %3D Is the origin an extreme point of conv S? Is the…

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Q: Prove that B (xX+1; y ) + B (x,y+1)=B (X,Y) let П (х+1) = ХГ (Х)

A: We have the formula for beta function interms of gamma function is that B(m,n)=Γ(m)Γ(n)Γ(m+n)As we…

Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Use Green Theorem to Evaluate |y x²dx – x²dy where C is left half of x² + y² = 25

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Q: Prove that if A C R and t > 0, then |A| = |AN(-t,t)|+|AN (R\(-t,t))|-

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Q: Prove that if X ⊆ Y then X ∩ Z ⊆ Y ∩ Z.

A: To prove this we will take an element from X∩Z and show that 'a' also belongs to Y∩Z.

Statements and Logic

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