Ordered pair

In the set theory, we learnt to write a set in different forms, we also learnt about different types of sets and studied operations on sets and Venn diagrams. Also in co-ordinate system we learnt about an ordered pair.

We studied ordered pair in co-ordinate system to locate a point. By the ordered pair (2, 5) we mean a pair of two integers, strictly in the order with 2 at first place called the abscissa and 5 at second place called the ordinate. 

The ordered pair (2, 5) is not equal to ordered pair (3, 2) i.e., (2, 5) ≠ (5, 2). Thus, in a pair, the order of elements is important. An ordered pair consists of two elements that are written in the fixed order. So, we define an ordered pair as: 

• The pair of elements that occur in particular order and are enclosed in brackets are called a set of ordered pairs. 

• If ‘a’ and ‘b’ are two elements, then the two different pairs are (a, b); (b, a) and (a, b); (b, a). 

• In an ordered pair (a, b), a is called the first component and b is called the second component. 

Suppose, if A and B are two sets such that a∈A and b∈B, then by the ordered pair of elements we mean (a, b) where 'a' is called the Iˢᵗ component and 'b' is called the IIⁿᵈ component of the ordered pair.

If the position of the components is changed, then the ordered pair is changed, i.e., it becomes (b, a) but (a, b) ≠ (b, a).

Note:

Ordered pair is not a set consisting of two elements.

Equality of Ordered Pairs:

Two ordered pairs are equal if and only if the corresponding first components are equal and corresponding second components are equal.

For example:

Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d, i.e., (a, b) = (c, d). Find the values of x and y, if (2x - 3, y + 1) = (x + 5, 7)

Solution:

By equality of ordered pairs, we have

2x - 3 = x + 5 and y + 1 = 7

⇒ 2x - x = 5 + 3 ⇒ x = 8 and y = 7 - 1 ⇒ y = 6

Note:

Both the elements of an ordered pair can be the same, i.e., (2, 2), (5, 5).

Two ordered pairs are equal if and only if the corresponding first components are equal and second components are equal.

For example:

1. Ordered pairs (x, y) and (2, 7) are equal if x = 2 and y = 7.

2. Given (x - 3, y + 2) = (4, 5), find x and y.

Solution:

(x - 3, y + 2) = (4, 5)

⇒ x - 3 = 4 and y + 2 = 5

Then x = 4 + 3 and y = 5 - 2 or x = 7 and y = 3

3. Given (3a, 3) = (5a - 4, b + 1)

Solution:

(3a, 3) = (5a - 4, b + 1)

Then, 3a = 5a - 4 and 3 = b + 1

⇒ 5a - 3a = 4 and b = 3 - 1

⇒ 2a = 4 and b = 2

⇒ a = 4/2

⇒ a = 2