Let "O" (addition) and "o" (multiplication) be two binary operations on the set V = {a | a € R} defined by Va, be V, ab= -ab and Vk eR, ko a = ka respectively. Then which of the following is/are always correct ? I. Visclosed under addition. II. V is closed under multiplication. III. V is not commutative with respect to addition. IV. The additive identity of V is -1. (a) Only I (b) Only II (c) Only II and IV (d) Only I, Il and IV (e) Only I, Il and III

Q: Given a set A = {0,1, z} equipped with addition + and multiplication · defined by the tables: + 0 1…

A: We can see that, Here, a×b=b×a. Therefore, Multiplication is commutative. Yes, is the required…

Q: Show that a permutation o E S, commutes with the transposition (a b) if and only we are in one of…

A: Given: f:S5→S5 is an automorphism. σ∈S5 To do: (d) Prove that σ commutes with the transposition ab…

Q: [1 F 3.11 |8 E 5||1 8 [4 _G 11 [1 4 1] 3 2] [4 2 1]

A: given A=01B=10C=11D=00 E=AT=01 F=BT=10 G=CT=11H=DT=00 K=101L=001M=KT=101N=LT=001

Q: (iv) The power set P(C) of C by listing down the elements directly.

A:

Q: State whether the following statements are true or false. Justify.(i) For an arbitrary binary…

A: i. Given,  For an arbitrary binary operation ∗ on a set N, a ∗ a = a ∀ a ∈ N.

Q: C. Let the binary operation * be defined on S = {a, b, c, d, e} by means of a given table. С. a b d…

A: Consider the table: ∗abcdeaabcdebbcaecccabbadbebededbadc From table: 1) Compute…

Q: 7. Consider the following table that displays the binary operation on the set (x, y, z). *…

A:

Q: 8) Define a set X recursively as follows: B) 30 e X R,) If xe X and x is even, then e X 2 R,) If xe…

A:

Q: Given a set A {0, 1, z} equipped with addition +and multiplication defined by the tables: 0 1 1 1 0.…

A: We have to find

Q: Let "O" (addition) and "o" (multiplication) be two binary operations on the set V = {a | a e R} %3D…

A:

Q: Let N = {0, 1,2, ...} and let R be a binary relation on the set Z = {(a, b) : a, b e N} defined by…

A: According to our guidelines we can do three subpart of a question please post remaining as a fresh…

Q: Let * be a binary operation on Z defined by a*b = a +b-4 for all a , beZ. Which of the following is…

A:

Q: Let "O" (addition) and "O" (multiplication) be two binary operations on the set V = {x|x E R,x > 0}…

A: Explanation of the answer is as follows

Q: x+y What is the identity element of x with respect to the binary operation • on R def ined by x •y=-…

A:

Q: 1. Let S = {a, b}. Define a binary operation ⋆ on S as follows: a⋆a=a a⋆b=a b⋆a=b b⋆b=b (a)…

A: "Since you have posted a question with multisubparts, we will solve first three parts for you. To…

Q: 2. Let + and · be the usual binary operations of addition and multiplication o the set Z. And let H…

A: Let, H be a non-empty set and • be a binary operation on H. Then H is said to be closed under binary…

Q: Suppose that G and H are equivalence relations in A, and that G C H. Prove that Gr Gy if and only if…

A: It is given that G and H are equivalence relations in A and G⊆H. The objective is to prove that Gx…

Q: For non empty binary relation R={(a, a).(a, b).(a, e).(b, b).(b, e).(c, C).(c, d).(d, d).(e, e)} on…

A:

Q: Let* be the binary operation on S =R x R defined, for all (a, b), (c,d) = S, by (a, b) * (c, d) = (a…

A: Given: S=ℝ×ℝ.The binary operation is defined by: for all a,b,c,d ∈ S,a,b*c,d=a+c, bc+ad

Q: Show that any language A is recognizable if and only if A <m ATM. Solution:

A: We have to show that any language A is recognizable if and only if A≤mATM.

Q: Let N be a nonempty set and o a permutation of N. Define a relation a~b if and only if b = o"(a) for…

A: Given, Ω is a nonempty set and σ a permutation of Ω. A relation on the set is defined as a~b if and…

Q: 7. Consider the following table that displays the binary operation on the set {x,y,z). I N 8 (a)…

A: This is a question from abstract algebra.

Q: Let * be a binary operation on R defined by a*b=a+b+2ab then find the identity element w.r.t. *. a.…

A: Given that a*b=a+b+2ab The objective is to find the identity. We say that e is identity element if…

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

A: Given below the detailed solution

Q: Let {0, 1}N = {(a, a1, a2, . ..) : an = 0 or 1, n E N}. • Construct a 1-1 correspondence between {0,…

A: let 0,1N=a0,a1,a2,....:an=0 or 1, n∈ℕ (i) construct a 1-1 correspondence between 0,1ℕ and all the…

Q: Let A = N × N and ∗ be the binary operation on A defined by(a, b) ∗ (c, d) = (a + c, b + d)…

A: Given:

Q: 2) Let m +0 be an integer. Show that Rm := { (a, b) E Z × Z|m divides a – b} is an equivalence…

A:

Q: Show that a permutation o E S; commutes with the transposition (a b) if and only we are in one of…

A: Hi! As per norms, we will be answering only question (d). If you need an answer to others then…

Q: Given that * :Zx Z Z is defined by m * n = m" for m, n e Z, then is an operation on the set of…

A:

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

A: our objective is to prove the result

Q: The following are equivalent in a Boolean algebra: (1)a+b=b, (2)a*b=a, (3)a'+b=1, (4)a*b'=0 i.…

A: The given equations are,1a+b=band2 ab=aWe need to show that these twp are equivalent.

Q: 2. Let the binary operation * defined on s = {a, b, c, d, e} by means of the table a. Compute the…

A: Given: Let the binary operation * defined on S=a,b,c,d,e by means of the table below.

Q: Select the following relations on Z that are equivalence relations. ○{(a,b) ||a| = |b|} {(a,b) a²b²…

A:

Q: 1. Which of the following binary relations defined on Z, the set of integers, Is HOI aRb e a is a…

A:

Q: 2. Let S be the set of three elements given by S = {A, B, C} with the following table. * A B A A B B…

A: From the given table , we can see that  A*B=B= B*AA*C=C=C*AB*C=A=C*B So, the binary operation * is…

Q: Let o be the 0= permutation 1 2 3 4 5 5 14 9 10 2 12 6 5 11 15 3 a) b) defined by: 6 7 6789 10 11 12…

A: Firstly change sigma in disjoint cycles and use the property to change cycle to transpositions.…

Q: 2. Prove thatN is the smallest inductive subset of R:

A: We can prove this by proving property of induction

Q: a) Let op be an associative operation withe as the neutral element: op is associative: (x op y) op z…

A: Here we have to prove three duality theorems. The given duality theorems are defined for fold…

Q: Let Σ = {0, 1, +, =} and S = {x = y+z : x,y,z ∈ {0,1}* and x,y,z are binary integers, and x is the…

A: Given that S=x=y+z By using the pumping lemma. Assume that for the purpose of reaching a…

Q: Express the following permutation as a product of disjoint cycles. ). 1 2 3 4 5 6 7 8 9 10 11 12 f 2…

A:

Q: let A to have0 al an must by

A:

Q: Supposing the binary operation * is defined on the set T = {1, 2, 3, 4, 5} by a * b = a +b+ 2ab. Say…

A:

Q: * is said to be commutative in A for all a,b ɛ A Select one: O a a*b b * a O b.a * b = b * a Oc a+b…

A: * is said to be commutative in A for all a, b in A, if, Option (b) a*b = b*a

Q: (b) The equivalence class of x € G is HxK coset of H and K. Show that the double cosets of H and K…

A: The given problem is to prove the following statements. We have to show that double cosets of H and…

Q: Decide whether the given set B is closed with respect to the binary operation defined on the set of…

A: Given: x*y=x+xy To find: whether the given set B is closed with respect to the binary operation…

Q: a) Let * be the operation on Z defined by a*b= a + b - 3ab for all a,b e Z. Justify the following…

A:

Q: Let f and g be any permutations. Show that the order of h=fgf^(-1) is the same as the order of g.

A:

Q: 6. Let * be the binary operation defined on Z by r * y = x + y +3. (a) Determine whether * is…

A:

Q: An operation * on a set S is a relationship that assigns to some ordered pair of elements of S a…

A: We have to tell about the statement is whether true or false