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: Let A = { 2, 3, 4 } and B = { 6, 8, 10, 12 } and define a relation R from A to B as follows: %3D For…

A:

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

A:

Q: 2. Determine whether the following pairs of operators commute: (a) D,. D.] (b) [D.. E] ( D.. A] (a)…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

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

A:

Q: Anikwenie 46. If P(A) = 0.5, P(B) = 0.40 and P(A U B) = 0.70, find P(A/B) and P(A U B) where A is…

A: Given that P(A) = 0.5 P(B) = 0.40 P(A U B) = 0.70

Q: Let o+ a eR and let G= {na| nez} ,then (G, +) is an abetian gronp. ( Hint: the identity element is…

A: Let 0≠a∈R and let G=na|n∈Z, then G, + is a group if the following properties are satisfied. 1.…

Q: Evaluate the deteminat of each matris. A= 5-1 [213 3. A-I0 2 20-2 4 A 6 0

A:

Q: Investigate if the following relationship 'is' or 'not' of order a ۲b ⟺ a ≤ b +1, a, b ∈ Z

A: A relation R on a set X is said to be order relation if it satisfies the following conditions,…

Q: C. Let T be defined onN N by (a, b)T (c, d) if and only if a c and b 2 d. Write down the full nature…

A: We will use the basic definition of reflexive, symmetric, antisymmetric and transitive realtion.

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

A:

Q: Let S = {1,2,3,4, 5}, G = {(1),(123), (132)}. For each element %3D a e S, find stabc(a) and orbg(a)

A:

Q: Use the adjoint of A = [1 [1 -5 -2 27to 1 to find A-¹ 0 2

A:

Q: give a clear and detailed proof of the following: if f: A implies B is a bijection, then f-1 : B…

A:

Q: {1,3, 5, 7, 9, 11}. Use the Let X = {4, 5, 6, 7, 8, 9} and Y roster method to describe each of the…

A:

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

A:

Q: ∃x ∈ D such that ∀y ∈ E, x + y = 1 Negation:

A: We need to find the negation of  ∃x ∈ D such that ∀y ∈ E, x + y=1

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

A:

Q: Let Y; and Yk be two principal components with associated 1; and Ak . Assume that Ak > lj. Which of…

A:

Q: [1 2 0] Let A = 1 2 0 0 1 0 and v = Select the statement that is correct. O v is an element of Col…

A: We have to find correct statement

Q: 24.4. Let A = {1, 2} and B = {3, 4). Write down all functions f : A → B. Indicate which are…

A: Given sets are    and                The cardinality of A is   (  Number of elements in A…

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

A:

Q: a. Write o using the two row form. b. Write o using cycle notation. c. Write the function o¬1 in the…

A:

Q: Let A = {0, {0}, {Ø, {Ø}}}. State whether each of the following statements is true or false: ØCA {0,…

A: Given A=ϕ,ϕ,ϕ,ϕ

Q: f(x, y) = e cos(ly). Find and classify all critical points of the function. If there are more blanks…

A: Note: According to Bartleby we have to answer only the first three subparts of the given question…

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: Prove if a e G, o(a) =n, then a = reN GCD(n,r)'

A: Given below the detailed solution

Q: (c) Using the set identities, prove that (A – B) n (A – C) = A – (B U C).

A: To Prove: A-B∩A-C=A-B∪C

Q: Let a, b, n E Z. Prove: If an and bln with gcd(a, b) = 1, then abln.

A: If g(a,b)=d then there exist r and s such that ar+bs=d. If a|c then it means there exists x such…

Q: Determine if the following statement is true or false. Motivate your answer. If Re(z) > Re(w) and…

A: We have to check whether the statements, "If Rez>Rew and Imz>Imw then z>w" Is true or…

Q: Reflexive. • Anti-reflexive. • Symmetric. • Anti-symmetric. Justify all your answers. Transitive. •…

A: Let's consider a few examples of elements inside the relation R, where (a,b)∈ R if and only if…

Q: Which one of the following statements is a true statement? O If f:A » B and g:B → C are functions…

A:

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…

Q: Show that ∗ : R × R → R given by           a ∗ b → a + 2b is not associative.

A: Given:Show that ∗: R × R → R given by           a ∗ b → a + 2b is not associative.

Q: Let -1 2 2 A = 1 -1 0 and U =|-2, then determine if %3D 3 -1 2 U is in Nul A. U is in Col A (i) (ii)

A: