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Discrete Mathematics Properties of Binary Operations

Discrete Mathematics Properties of Binary Operations with introduction, sets theory, types of sets, set operations, algebra of sets, multisets, induction, relations, functions and algorithms etc.

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Properties of Binary Operations

There are many properties of the binary operations which are as follows:

1. Closure Property: Consider a non-empty set A and a binary operation * on A. Then is closed under the operation *, if a * b ∈ A, where a and b are elements of A.

Example1: The operation of addition on the set of integers is a closed operation.

Example2: Consider the set A = {-1, 0, 1}. Determine whether A is closed under

  1. Addition
  2. Multiplication

Solution:

(i)The sum of elements is (-1) + (-1) = -2 and 1+1=2 does not belong to A. Hence A is not closed under addition.

(ii) The multiplication of every two elements of the set are

              -1 * 0 = 0;         -1 * 1 =-1; -1 * -1 = 1
              0 * -1 = 0;         0 * 1 = 0; 0 * 0 = 0
              1 * -1 = -1;         1 * 0 = 0; 1 * 1 = 1

Since, each multiplication belongs to A hence A is closed under multiplication.

2. Associative Property: Consider a non-empty set A and a binary operation * on A. Then the operation * on A is associative, if for every a, b, c, ∈ A, we have (a * b) * c = a* (b*c).

Example: Consider the binary operation * on Q, the set of rational numbers, defined by a * b = a + b - ab ∀ a, b ∈ Q.

Determine whether * is associative.

Solution: Let us assume some elements a, b, c ∈ Q, then the definition

              (a*b) * c = (a + b- ab) * c = (a + b- ab) + c - (a + b- ab)c
                      = a + b- ab + c - ca -bc + abc = a + b + c - ab - ac -bc + abc.

Similarly, we have
              a * (b * c) = a + b + c - ab - ac -bc + abc

Therefore,         (a * b) * c = a * (b * c)

Hence, * is associative.

3. Commutative Property: Consider a non-empty set A,and a binary operation * on A. Then the operation * on A is associative, if for every a, b, ∈ A, we have a * b = b * a.

Example: Consider the binary operation * on Q, the set of rational numbers, defined by a * b = a2+b2 ∀ a,b∈Q.

Determine whether * is commutative.

Solution: Let us assume some elements a, b, ∈ Q, then definition

              a * b = a2+b2=b * a

Hence, * is commutative.

4. Identity: Consider a non-empty set A, and a binary operation * on A. Then the operation * has an identity property if there exists an element e in A such that a * e (right identity) = e * a (left identity) = a ∀ a ∈ A.

Example: Consider the binary operation * on I+, the set of positive integers defined by a * b =Discrete Mathematics Properties of Binary Operations

Determine the identity for the binary operation *, if exists.

Solution: Let us assume that e be a +ve integer number, then

              e * a, a ∈ I+
              Discrete Mathematics Properties of Binary Operations= a, e = 2...............equation (i)

Similarly,         a * e = a, a ∈ I+
              Discrete Mathematics Properties of Binary Operations=2 or e=2...........equation (ii)

From equation (i) and (ii) for e = 2, we have e * a = a * e = a

Therefore, 2 is the identity elements for *.

5. Inverse: Consider a non-empty set A, and a binary operation * on A. Then the operation is the inverse property, if for each a ∈A,,there exists an element b in A such that a * b (right inverse) = b * a (left inverse) = e, where b is called an inverse of a.

6. Idempotent: Consider a non-empty set A, and a binary operation * on A. Then the operation * has the idempotent property, if for each a ∈A, we have a * a = a ∀ a ∈A

7. Distributivity: Consider a non-empty set A, and a binary operation * on A. Then the operation * distributes over +, if for every a, b, c ∈A, we have
                            a * (b + c) = (a * b) + (a * c)         [left distributivity]
                            (b + c) * a = (b * a) + (c * a)         [right distributivity]

8. Cancellation: Consider a non-empty set A, and a binary operation * on A. Then the operation * has the cancellation property, if for every a, b, c ∈A,we have
                            a * b = a * c ⇒ b = c         [left cancellation]
                            b * a = c * a ⇒ b = c         [Right cancellation]


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