Answer to Question #284617 in Abstract Algebra for Abdullah

Question

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

Answer 1

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