Let A=N×N and * be the binary operation on A defined by (a, b)*(c, d)=(a+c,b+d) Show that * is commutative and associative. Find the identity element for * on A, if any.
Let A=N×N and * be the binary operation on A defined by (a, b)*(c, d)=(a+c,b+d) Show that * is commutative and associative. Find the identity element for * on A, if any.
Solution and Explanation
A = N × N
* is a binary operation on A and is defined by:
(a, b) * (c, d) = (a + c, b + d)
Let (a, b), (c, d) ∈ A
Then, a, b, c, d ∈ N
We have:
(a, b) * (c, d) = (a + c, b + d)
(c, d) * (a, b) = (c + a, d + b) = (a + c, b + d)
[Addition is commutative in the set of natural numbers]
∴ (a, b) * (c, d) = (c, d) * (a, b)
Therefore, the operation * is commutative.
Now, let (a, b), (c, d), (e, f) ∈A
Then, a, b, c, d, e, f ∈ N We have:
((a,b) * (c,d)) * (e,f) = (a+c,b+d) * (e,f) = (a+c+e,b+d+f)
(a,b) * ((c,d) * (e,f))= (a,b) * (c+e,d+f) = (a+c+e,b+d+f)
∴ ((a,b) * c,d)) * (e,f)= (a,b) * ((c,d) * (e,f))
Therefore, the operation * is associative.
An element e=(e1,e2)∈A will be an identity element for the operation * if a*e=a=e*a∀ a=(a1,a2)∈A,i.e,(a1+e1,a2+e2)=(a1,a2)=(e1+a1,e2+a2),which is not true for any element in A.
Therefore, the operation * does not have any identity element.
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Concepts Used:
Binary Operation
A binary operation can be understood as a function f (x, y) that applies to two elements of the same set S, such that the result will also be an element of the set S. Examples of binary operations are the addition of integers, multiplication of whole numbers, etc. A binary operation is a rule that is applied on two elements of a set and the resultant element also belongs to the same set.
Properties of Binary Operation:
- Closure Property: A binary operation * on a non-empty set P has closure property, if a ∈ P, b ∈ P ⇒ a * b ∈ P.
- Associative Property: The associative property of binary operations holds if, for a non-empty set S, we can write (a * b) *c = a*(b * c), where {a, b, c} ∈ S. Commutative Property: A binary operation * on a non-empty set S is commutative, if a * b = b * a, for all (a, b) ∈ S. Suppose addition be the binary operation and N be the set of natural numbers.
- Distributive Property: Let * and # be two binary operations defined on a non-empty set S. The binary operations are distributive if, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ S. Suppose * is the multiplication operation and # is the subtraction operation defined on Z (set of integers).
- Identity Element: A non-empty set P with a binary operation * is said to have an identity e ∈ P, if e*a = a*e= a, ∀ a ∈ P. Here, e is the identity element.
- Inverse Property: A non-empty set P with a binary operation * is said to have an inverse element, if a * b = b * a = e, ∀ {a, b, e}∈P. Here, a is the inverse of b, b is the inverse of a and e is the identity element.
Read More: Truth Table
Types of Binary Operation:
There are four main types of binary operations which are: