Define * on Z by a * b = a + b – ab. Show that * is a binary operati

1.	Define * on Z by a * b = a + b – ab. Show that * is a binary operation on Z which is commutative as well as associative.
Answer» Consider a, b ∈ Z where a * b = a + b – ab and b * a = b + a – ba So we get a * b = b * a Associative: Consider a, b, c ∈ Z Here, (a * b) * c = (a + b – ab) * c = a + b – ab + c – (a + b – ab) c We get (a * b) * c = a + b + c – ab – bc – ca + abc a * (b * c) = a * (b + c – bc) = a + b + c – bc – a (b + c – bc) We get a * (b * c) = a + b + c – an – bc – ca + abc So (a * b) * c = a * (b * c) Therefore, operation on Z is associative.

Define * on Z by a * b = a + b – ab. Show that * is a binary operation on Z which is commutative as well as associative.

Answer»

Consider a, b ∈ Z where a * b = a + b – ab and b * a = b + a – ba

So we get a * b = b * a

Associative:

Consider a, b, c ∈ Z

Here,

(a * b) * c = (a + b – ab) * c = a + b – ab + c – (a + b – ab) c

We get

(a * b) * c = a + b + c – ab – bc – ca + abc

a * (b * c) = a * (b + c – bc) = a + b + c – bc – a (b + c – bc)

We get

a * (b * c) = a + b + c – an – bc – ca + abc

So (a * b) * c = a * (b * c)

Therefore, operation on Z is associative.

Discussion

No Comment Found

Related InterviewSolutions

Determine whether each of the following operations define a binary operation on the given set or not: ‘*’ on N defined by a * b = ab for all a,b N.
Determine whether each of the following operations define a binary operation on the given set or not: ‘O’ on Z defined by a O b = ab for all a,b Z.
Determine whether each of the following operations define a binary operation on the given set or not: ‘x6’ on S={1,2,3,4,5} defined by a x6 b = Remainder when ab is divided by 6.
On R – {1}, a binary operation * is defined by a * b = a + b – ab. Prove that * is commutative and associative. Find the identity element for * on R – {1}. Also, prove that every element of R – {1} is invertible.
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this. On Z + , defined * by a*b = |a – b| Here, Z + denotes the set of all non – negative integers.
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this. On Z+ , defined * by a*b = ab Here, Z+ denotes the set of all non – negative integers.
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this.On R, defined * by a*b = ab2Here, Z+ denotes the set of all non – negative integers.
Let * be a binary operation on the set Q of all rational numbers given as a * b = (2a – b)2 for all a, b ∈ Q. Find 3 * 5 and 5 * 3. Is 3 * 5 = 5 * 3?
Let S be the set of all real numbers except – 1 and let ‘*’ be an operation defined by a*b = a + b + ab for all ab∈S. Determine whether ‘*’ is a binary operation on ‘S’. if yes, Check its commutativity and associativity. Also, solve the equation (2*x)*3 = 7.
The binary operation * defined on R×R → R is defined as a*b = 2a + b. Find (2*3)*4.

Define * on Z by a * b = a + b – ab. Show that * is a binary operation on Z which is commutative as well as associative.

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment