Properties of Binary OperationsThere are many properties of the binary operations which are as follows: 1. Closure Property: Consider a nonempty 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
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 = 10 * 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 nonempty 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 Therefore, (a * b) * c = a * (b * c) Hence, * is associative. 3. Commutative Property: Consider a nonempty 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 = a^{2}+b^{2} ∀ a,b∈Q. Determine whether * is commutative. Solution: Let us assume some elements a, b, ∈ Q, then definition a * b = a^{2}+b^{2}=b * aHence, * is commutative. 4. Identity: Consider a nonempty 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 = 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_{+} Similarly, a * e = a, a ∈ I_{+} 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 nonempty 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 nonempty 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 nonempty set A, and a binary operation * on A. Then the operation * distributes over +, if for every a, b, c ∈A, we have 8. Cancellation: Consider a nonempty set A, and a binary operation * on A. Then the operation * has the cancellation property, if for every a, b, c ∈A,we have
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