Find the identity element for the following binary operators defined on the set Z. If identity element exists then find the inverse element also. a. a eb = a + b+ ab b. a e b = a + b - 5
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- Label each of the following statements as either true or false. 1. Mapping composition is a commutative operation.Label each of the following statements as either true or false. Let ={a,b,c}. The power set P(A) is closed with respect to the binary operation of forming intersections.Define the binary operator ⊗⊗ by: a⊗b=a2+b+5a⊗b=a2+b+5 Find each of the following: 7⊗4=7⊗4= 2⊗2=2⊗2= 4⊗7=4⊗7= s⊗r=s⊗r=
- Define the binary operator # by: aa#b=b= the larger value of aa or bb.; Find each of the following: 99#3=3= 88#8=8= 33#9=9=Define the binary operator ∇∇ by: a∇b=5a∇b=5 Simplify each of the following. Do the order of operations (do what is in parentheses first). (5∇4)∇2(5∇4)∇2 = (6∇3)∇7(6∇3)∇7Let 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.
- 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 = aLet S be the set of four elements given by S = {A, B, C, D} with the following table.* A B c D. If there is an identity element, which elements have inverses?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. Prove the equivalence of (1) and (2) .