The binary operation ab=a+b-3ab is defined on Q. Select the correct statements. Select the correct statements. O The binary operation is associative. The bibary operation is closed.
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- Prove that a nonzero element in is a zero divisor if and only if and are not relatively prime.19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.Define an operation ∗ on Z+ as follows: For a,b ∈ Z+, a∗b = a−b+1. Show that ∗ is a binary operation on Z. What is 3 ∗ 5? Is ∗ commutative? Is (2 ∗ 3) ∗ 4 = 2∗(3∗4)? Can you conclude that is ∗ associative
- On R we define the following binary operation; x*y = xy + 2ax + by Find a and b such that * is commutative and associative.Let A = N × N and ∗ be the binary operation on A defined by(a, b) ∗ (c, d) = (a + c, b + d) Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).
- Determine which elements of Z8 have inverses under the op-eration ×8. Then do the same for Z7 under ×7.Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.(i) On Z+, define ∗ by a ∗ b = a – b(ii) On Z+, define ∗ by a ∗ b = ab(iii) On R, define ∗ by a ∗ b = ab2(iv) On Z+ , define ∗ by a ∗ b = | a – b |(v) On Z+, define ∗ by a ∗ b = aProve that √2 is not an element of Q(π ).
- In each part following, a rule that determines a binary operation * on the set Z of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element. x * y = xY for x, y E Z +Suppose a is divisible by b. Show that gcd(a, b) = b.In each part following, a rule that determines a binary operation * on the set Z of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element. a.x*y=x+xy