Let S = R−{−1}. Define an operation ∗ on S by a∗b = a+b+ab, a,b ∈ S. (6.1) Show that S is closed under the operation ∗. (6.2) What is the identity in S under ∗? (6.3) What is the inverse of a ∈ S under ∗? (6.4) Assuming that ∗ is associative, show that (S,∗) is an abelian group. (6.5) Solve for x in the equation 1 ∗ x = 2.

Q: 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

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Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: Find the Boolean product AOB where A and B are defined below. ΓΟ 11 [1 A = 1 LO 1 0 11 001 0 0 1 and…

A: The Boolean matrices have entries consisting of only 0's and 1's. The Boolean matrix product can be…

Q: Simplify the following functional expression using Boolean algebra and its identities. List the…

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Q: 6. Prove that the adjoint of the adjoint is the original operator, that is, that L = is from section…

A: Let us prove the above result.

Q: Let 2 = {a, b, c}. Write down all possible o-algebras on N.

A: Given: Ω=a,b,c

Q: 1. Let T be a group of all invertible 2 x2 matrices of the form Where a.b.C ER and ac+0. Let / be…

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Q: Which of the following operation on Z is commutative Select one: a. x *y x + xy b. x *y 3(x + y) %3D…

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Q: True or false. If true, prove it; if false, provide a counter example. If E ⊂ [a, b], then a ≤ inf…

A: We have to prove that If E ⊂ [a, b], then a ≤ inf E ≤ sup E ≤ b.

Q: Prove the following set identities: (a) (AUB) nC = (ANC) U (BNC); (b) (AUB) - C= (A-C) U (B-C); (c)…

A: The prove of the identities (a), (b) and (c) are given below.

Q: Question 3 Take a look at the operator *, defined as follows: *: ZxZ→Z x*y = x + 3y (a.) Is the…

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Q: R. K Which of the following is sufficient to prove that A IJK is similar to A RST? A. ZI ZR and ZK…

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Q: How many possible Jordan canonical forms are there for a linear operator T: C6 C6 with…

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Q: Determine all homomorphisms from D5 onto Z₂ Z₂. Determine all homomorphisms from D5 to Z₂ + Z₂.

A: We have to determine all homomorphism from  onto .We also have to find homomorphism from  to

Q: How do you derermine of RS → R* 76 onto?

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Q: Let A = {1, 2, 3, 4, 5). Define an operation v by a vb= max {a, b}. Prepare its composition table.…

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Q: work): Evaluate the det(A) using any method (but make sure to show your A -2 3 -12 12 -14 19 -20 31…

A: Given:The matrix is given by .To Find:We have to find the determinant of the matrix .

Q: 14. Prove that AC B if and only if An BC= 0.

A: The objective is to prove the above question.

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

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Q: e multiplicative inverse of 1+ 3x in Z,[x] is: 4x+1 None 1/(3x+1) 6x+1

A: Consider the given information: To find multiplicative inverse of  1+3x in ℤ9x.

Q: Let A = {9k +7 ke Z} and B = {3j+1 je Z}. Show that AC B. :

A: Let, x ∈ A    => x = 9k + 7 for some k ∈ Z.

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

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Q: C Compute the following expressions (d) Im a+bj a+cj Im [+

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Q: (b) Let A and Bị be independent; similarly, let A and B2 be independent. Show that in this case, A…

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Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: Q1-(Uniqueness of the inverse:). Show that for every a ∈ G, the inverse of a is unique. We will…

A: We need to show that for every a ∈ G, the inverse of a is unique.To show this follow below steps:…

Q: Let G = Z5OU(7). The identity of G is O (0, 1) O (1, 1) O None of the choices O (1, 0) O (0, 0)

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Q: Answer the following questions (Q1) (a) If * be a binary operation on Z given by x * y x. Determine…

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Q: n=1 Let {En} be a countable collection of Lebesgue measurable ibset of R. Suppose E is a nonempty…

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Q: Find and draw Ubsolb, c0] × [-ab, ab] and Uaso Ubsolb, o0] × [-ab, ab] and Naso Ubsolb, o0] × [-ab,…

A: 1. ∪b>0b,∞×-ab,ab Here a is fixed. Fix a=2. And fix b=3 to understand what b,∞×-ab,ab represents…

Q: Task 1.5 For let a,b E Z = Z \ {0} ' GCD (a,b) denote the greatest common divisor of a and b. The…

A: COMMUTATIVE PROPERTY Consider the values, a=8b=12 Consider the following, GCD8,12=4GCD12,8=4 Since…