Relations and Functions

Binary Operations

We are quite familiar with arithmetic operations like addition, subtraction, division, and multiplication. Also, we know about exponential function, log function etc. Today we will learn about the binary operations. As the name suggests, binary stands for two. Does that mean that we can use two functions simultaneously using binary operation? Let’s find out.

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Binary Operation

Just as we get a number when two numbers are either added or subtracted or multiplied or are divided. The binary operations associate any two elements of a set. The resultant of the two are in the same set. Binary operations on a set are calculations that combine two elements of the set (called operands) 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. It is an operation of two elements of the set whose domains and co-domain are in the same set.

binary operation

Addition, subtraction, multiplication, division, exponential is some of the binary operations.

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Properties of Binary Operation

  • Closure property: An operation * on a non-empty set A has closure property, if a ∈ A, b ∈ A ⇒ a * b ∈ A.
  • Additions are the binary operations on each of the sets of Natural numbers (N), Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

The additions on the set of all irrational numbers are not the binary operations.

  • Multiplication is a binary operation on each of the sets of Natural numbers (N), Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

Multiplication on the set of all irrational numbers is not a binary operation.

  • Subtraction is a binary operation on each of the sets of Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

Subtraction is not a binary operation on the set of Natural numbers (N).

  • A division is not a binary operation on the set of Natural numbers (N), integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).
  • Exponential operation (x, y) → xy is a binary operation on the set of Natural numbers (N) and not on the set of Integers (Z).

Types of Binary Operations

Commutative

A binary operation * on a set A is commutative if a * b = b * a, for all (a, b) ∈ A (non-empty set). Let addition be the operating binary operation for a = 8 and b = 9, a + b = 17 = b + a.

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Associative

The associative property of binary operations hold if, for a non-empty set A, we can write (a * b) *c = a*(b * c). Suppose N be the set of natural numbers and multiplication be the binary operation. Let a = 4, b = 5 c = 6. We can write (a × b) × c = 120 = a × (b × c).

Distributive

Let * and o 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). Consider * to be multiplication and o be subtraction. And a = 2, b = 5, c = 4. Then, a*(b o c) = a × (b − c) = 2 × (5 − 4) = 2. And (a * b) o (a * c) =  (a × b) − (a × c) = (2 × 5) − (2 × 4) = 10 − 6 = 2.

Identity

If A be the non-empty set and * be the binary operation on A. An element e is the identity element of a ∈ A, if a * e = a = e * a. If the binary operation is addition(+), e = 0 and for * is multiplication(×), e = 1.

Inverse

If a binary operation * on a set A which satisfies a * b = b * a = e, for all a, b ∈ A. a-1 is invertible if for a * b = b * a= e, a-1 = b. 1 is invertible when * is multiplication.

Solved Example for You

Question 1: Show that division is not a binary operation in N nor subtraction in N.

Answer : Let a, b ∈ N

Case 1: Binary operation * = division(÷)

: N × N→N given by (a, b)  (a/b) ∉ N (as 5/3 ∉ N)

Case 2: Binary operation * = Subtraction(−)

: N × N→N given by (a, b) a − b ∉ N (as 3 − 2 = 1 N but 2−3 = −1 ∉ N).

Question 2: Are all binary operations closed?

Answer: Many sets that you might be familiar to are closed under certain binary operators, whereas many are not. Thus, the set of odd integers remains closed under multiplication. For instance, the set of odd integers is not closed under addition, as the sum of two odd numbers is not always odd, actually, it is never odd.

Question 3: Is square root a binary operation?

Answer: A non-binary operation refers to a mathematical process which only requires one number to achieve something. Addition, subtraction, multiplication, and division are examples of binary operations. Similarly, examples of non-binary operations consist of square roots, factorials, as well as absolute values.

Question 4: What is the identity element in a binary operation?

Answer: An identity element or neutral element in binary operation refers to a special kind of element of a set with regards to a binary operation on that set, that leaves an element of the set unaffected when combined with it. We use this concept in algebraic structures like groups and rings.

Question 5: What is the binary overflow?

Answer: Overflow takes place when the magnitude of a number surpasses the range permitted by the size of the bit field. The sum of two identically-signed numbers may very well surpass the range of the bit field of those two numbers, and thus overflow may be a possibility in this case.

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Al quba
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Al quba

The example for onto function doesn’t qualify as a function in the first place. Does it??
It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??
Please tell me if I’m correct or not.
It is really confusing.

Eli Simeon
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Eli Simeon

An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if… Read more »

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