Question #284617

7. For each operation * given below, determine whether * is a binary operation, commutative or associative. In the event that * is not a binary operation, give justification for this.



i. On Z, a*b=a-b



ii. On Q, a*b=ab+1



iii. On Q, a*b=ab/2



iv. On Z+ , a*b= 2ab




1
Expert's answer
2022-01-04T16:19:07-0500

binary operation is a rule for combining two elements (called operands) to produce another element

Property of Binary Operation is closure:

if aA,bA    abAa\isin A,b\isin A\implies a*b\isin A

operation is commutative if

ab=baa*b=b*a

operation is associative if

(ab)c=a(bc)(a*b)*c=a*(b*c)


i.

binary operation

not commutative: abbaa-b\neq b-a

not associative: (ab)ca(bc)=(ab)+c(a-b)-c\neq a-(b-c)=(a-b)+c


ii.

binary operation

commutative: ab+1=ba+1ab+1=ba+1

associative: (ab)c+1=a(bc)+1(ab)c+1=a(bc)+1


iii.

binary operation

commutative: ab/2=ba/2ab/2=ba/2

associative: (ab)c/2=a(bc)/2(ab)c/2=a(bc)/2


iv.

binary operation

commutative: 2ab=2ba2ab=2ba

associative: 2(ab)c=2a(bc)2(ab)c=2a(bc)


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