Answer to Question #284617 in Abstract Algebra for Abdullah

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 "a\\isin A,b\\isin A\\implies a*b\\isin A"

operation is commutative if

"a*b=b*a"

operation is associative if

"(a*b)*c=a*(b*c)"


i.

binary operation

not commutative: "a-b\\neq b-a"

not associative: "(a-b)-c\\neq a-(b-c)=(a-b)+c"


ii.

binary operation

commutative: "ab+1=ba+1"

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


iii.

binary operation

commutative: "ab\/2=ba\/2"

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


iv.

binary operation

commutative: "2ab=2ba"

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


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