Binary Operation: Overview, Questions, Preparation

Binary operations are operations dealing with two operands, hence the name “binary”. This concept explores the idea that no matter how many operands exist, only two can be solved at a time. The operations this works on are “addition, subtraction, division and multiplication.”

In terms of sets, the binary operations associate any two elements of a set. The resultant of the two will be in the same set. Binary operations on a set are calculations that combine two set elements to produce another element of the same set.

The binary operations * on a non-empty set A are functions from A × A to A. The binary operation, *: A × A → A. So the operation * performed on operands a and b is denoted by a * b.

Types of Binary Functions

There are four main types of binary operations which are:
1. Binary Addition
+ : R x R → R is given by (a, b) → a + b
2. Binary Subtraction
–: R x R → R is given by (a, b)→ a – b
3. Binary Multiplication
x: R x R → R is given by (a, b) → a x b
4. Binary Division
: R x R → R is given by (a, b) → a  b

The domain and codomains are fundamental to be remembered. Because in addition and multiplication, it can apply to natural numbers thus we can ever write N x N → N... Still, this may not end up in the domain of the natural numbers for division and subtraction.

Properties of Binary Functions

1. Closure property
An operation * on a non-empty set A has closure property, if a ∈ A, b ∈ A ⇒ a * b ∈ A.

2. Commutative property
* on a set A is commutative if a * b = b * a, for all (a, b) ∈ A (non-empty set).

3. Associative property

For a non-empty set A, we can write (a * b) *c = a*(b * c).

4. Distributive property

Let * and to be two binary operations defined on a non-empty set A. The binary operations are distributive if a*(b o c) = (a * b) o (a * c) or (b o c)*a = (b * a) o (c * a).

5. Identity property
* on set A, if a * e = a = e * a, it has identity property.

6. Inverse property
* on a set A which satisfies a * b = b * a = e, for all a, b ∈ A.

Students are introduced to this topic in Class 12 to understand the mathematical representation of all operands for a function. This has around four marks weightage in the Class 12 exams and is not incorporated much in the entrance examinations. But this is important to get a detailed explanation of further aspects of the relations and functions chapter.

1. Show that division and subtraction is not a binary operation in the set N.
Solution.

Let a, b ∈ N
For * = division(÷)
–: N × N→N given by (a, b) → (a/b) ∉ N (as 5/3 ∉ N)
 For * = Subtraction(−)
–: N × N→N given by (a, b)→ a − b ∉ N (as 3 − 2 = 1 ∈ N but 2−3 = −1 ∉ N).

2. Is square root a binary operation?
Solution.

No. A non-binary operation requires one number.

3.What is the binary overflow?
Solution.

Overflow takes place when the magnitude of a number surpasses the range permitted by the set.

Q: What is an identity element for addition?

Q: Is multiplication an inverse property?

Q: Give an example of a non-binary function?

Q: Is exponential function binary?

Q: 1 is the identity element for?