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High School Mathematics - 2
5.5 Binary Operations

A binary operation is an operation that operates on two operands. In other words, a binary operation on anon-empty set A takes two elements of the set A and combines them to form an element of A. For instance '+' and 'x' are binary operations on the set l of integers. These operations are binary because only two integers are involved. The prefix"bi" denotes two a. The operations of union and intersection of sets and composition of functions are also binary operations.These operations mentioned here are denoted as follows:
a+b = c, a.b = c, A U B = C, gof = h
In each situation, an element (c, C or h) is assigned to an original pair of elements.

Definition: A binary operation on a set S is a mapping S X S into S.

Steps to check if an operation is binary or not

  1. Addition is a binary operation on the set of even natural numbers (the sum of two even natural numbers is an even natural number) but is not a binary operation on the set of odd natural numbers.(the sum of two odd natural numbers is a even natural number)
  2. Extracting the square root is a unary operation in arithmetic since in this case only one element in our number system is operated upon.
  3. neither addition nor multiplication is a binary operation on s = {0,1,2,3,4} because if we consider 2+3 = 5 does not belong to S and 2x3 = 6 does not belong to S.
Example: Check if the following operation is binary on I(set of integers)
a*b = a/b, a, b I
Consider a = 1, b = 2
a* b = a/b
a/b = 1/2 does not belong to I
Hence not a binary operation.

Directions: Check whether the following operations are binary.
Q 1: S = {2,4,6,8....} under addition.

Q 2: S = {0,1,2,3,4} under addition

Q 3: a*b = [(x2 + y2)] on I

Q 4: S = {1,3,5,7....} under addition

Q 5: S = {0,1} under addition.

Q 6: S = {1,3,5,7....} under mulitplication.

Q 7: S = {0,1} under multiplication.

Q 8: a*b = [(x2 + y2)] on Q(rational numbers)

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Question 10: This question is available to subscribers only!

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