Binary operation * is defined on set Z as a*b =a2
+b2
+ab+2 then 3*4= .................
Answers
Answer:(i) a∗b=a−b
Check commutative is
a∗b=b∗a
a∗b=a−b
b∗a=b−a
Since, a∗b
=b∗a
∗ is not commutative.
Check associative
∗ is associative if
(a∗b)∗c=a∗(b∗c)
(a∗b)∗c=(a−b)
∗
c=(a−b)−c=a−b−c
a∗(b∗c)=a∗(b−c)=a−(b−c)=a−b+c
Since (a∗b)∗c
=a∗(b∗c)
∗ is not an associative binary operation.
(ii) a∗b=a
2
+b
2
Check commutative
∗ is commutative if a∗b=b∗a
a∗b=a
2
+b
2
b∗a=b
2
+a
2
=a
2
+b
2
Since a∗b=b∗a∀a,bϵQ
∗ is commutative.
Check associative
∗ is associative if
(a∗b)∗c=a∗(b∗c)
(a∗b)∗c=(a
2
+b
2
)∗c=(a
2
+b
2
)
2
+c
2
a∗(b∗c)=a∗(b
2
+c
2
)=a
2
+(b
2
+c
2
)
2
Since (a∗b)∗c
=a∗(b∗c)
∗ is not an associative binary operation.
(iii) a∗b=a+b
Check commutative
∗ is commutative is a∗b=b∗a
a∗b=a+ab;b∗a=b+ba
Since a∗b
=b∗a
∗ is not commutative.
(iv) a∗b=(a−b)
2
Check commutative
∗ is commutative if a∗b=b∗a
a∗b=(a−b)
2
;b∗a=(b−a)
2
=(a−b)
2
Since a∗b=b∗a∀a,bϵQ
∗ is commutative.
Check associative
∗ if
(a∗b)∗c=a∗(b∗c)
(a∗b)∗c=(a−b)
2
∗c=[(a−b)
2
−c]
2
a∗(b∗c)=a∗(b−c)
2
=[a−(b−c)
2
]
2
Since (a∗b)∗c
=a∗(b∗c)
∗ is not an associative binary operation.
(v) a∗b=
4
ab
Check commutative.
∗ is commutative if a∗b=b∗a
a∗b=
4
ab
;b∗a=
4
ba
=
4
ab
Since a∗b=b∗a∀a,bϵQ
∗ is commutative.
Check associative.
∗ is association if (a∗b)∗c=a∗(b∗c)
(a∗b)∗c=(
4
4
ab
∗c
)=
16
abc
a∗(b∗c)=a∗(
4
bc
)=
4
a×
4
bc
=
16
abc
Since (a∗b)∗c=a∗(b∗c)∀a,b,cϵQ
∗ is an associative binary operation.
(vi) a∗b=ab
2
check commutative.
∗ is commutative if a∗b=b∗a
a∗b=ab
2
;b∗a=ba
2
Since a∗b
=b∗a
∗ is not commutative.
Check associative
∗ is associative if (a∗b)∗c=a∗(b∗c)
(a∗b)∗c=ab
2
∗c=(ab
2
)c
2
=ab
2
c
2
.
a∗(b∗c)=a∗bc
2
=a(bc
2
)
2
=ab
2
c
4
Since (a∗b)∗c
=a∗(b∗c)
∗ is not an associate binary operation.
Step-by-step explanation: