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

Q: . Rewrite the equality of ideals (r+ y/2) = (3 – 4/2) in Z[/2] in terms of elements.

A: We know that, A subring  B of a ring R is called a ideal of R if for every r∈R and for every b∈B,…

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

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Q: Suppose that j and k are positive integers such that j< k. Give a combinatorial proof of t identity:…

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Q: Use the element method and contradiction to prove that for all sets S and T S Ռ ( T - S) = {  }

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Q: The set G = :a is closed under addition %3D 0 2a

A: Use the closure property of addition.

Q: In R[x], prove ideals (x3 – 1, x7 – 1) = (x – 1, x – 1).

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

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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: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

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

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Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: let r ∈ R\ Q. prove that for all a,b,c ∈ Q, a+br=cr iff a=0 and b=c.

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Q: Let A & M and μ*(A) < oo. st B₁, B2,..., BEC, k <co with Bin B „²(AAŬB₂) <« j=1 any two sets E₁ and…

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Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find Z_2×Z_2.

Q: Show that is isomorphic to Z2.

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

A: Given information: It is given that X is a full rank matrix.

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

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

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Q: Suppose  v,w is an element of V. Explain why there exists a unique x which is an element of V such…

A: Suppose v, w is an element of V. Explain why there exists a unique x which is an element of V such…

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 P(Ø) = 0. Hint: Let A, = Ø for all i in equation (1.3.3).

A: Consider that

Q: Find the value(s) of k such that A is singular.

A: A matrix is singular if its determinant value is zero. So we find the determinant value of the given…

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

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: For each subfield S of Q[i, z], determine Auts(Q[i, z]).

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Q: Let S = {n + m√2 : n, m ∈ Z}  To prove that the sum of any two elements of S is an element of S.…

A: Given: S=n+m√2:n,m∈Z  (a) Let a+b2∈S for all a,b∈Z and let c+d2∈S for all c,d∈Z Add both these…

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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Q: Let a = 8 and b = 9. Find m,n E Z such that am+bn =1.

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Q: 4. ber A be set o and let a a: P(A)→P(A) be definec by 2 (x ) = A - X %3D is a bijection. Show that…

A: Let A be a set and let PA be the power set of A. Define a map α:PA→PA by: αx=A-x for all x∈PA…

Q: Represent algebraically and graphically the set of values of z for which |z+3|=|z−2i|

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Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

A: By adding the cubes of positive integers we solve the given problem.

Q: Suppose v; w is the element of V . Explain why there exists a unique x is the element of V such…

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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: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: c If in is an involution of X, show that (r, y)" = (yi", xi") is an involution of X2). %3D

A: Given X is a lattice. Given in  is an involution of X To show: ((x,y)in=(yin,xin) is an involution…

Q: 7. Prove there exists a unique x E R such that for every y € R, ту + т — 2 3D 2у.

A: We have to find the unique value of x for which the given equation is valid. We have, xy + x -2= 2y…

Q: 6. Let K be compact in R and let c E R. Prove that there exists a point a € K such that |c – a| =…

A: Given- Let K be compact in R and let c ∈ R.  To Prove- Prove that there exists a point a ∈ K such…

Q: For any x E R, there exists n EN such that x < n; ylgn ihi

A: We have to state whether given statement is true or false.

Q: Let 5= [(a,0); VER] show that s is subring subring-ft

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Q: Find A' for the given choice of U. If A={positive integers}, U=Z

A: We want to find  A' for the given choice of U. If A={positive integers}, U=Z

Q: Represent in the Argand Diagram all such z for which |z+ 15 – 9i| <1

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Q: For each xe X and a nbd: U(x) of x there exists another nbd.

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Q: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

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: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

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Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!