Simply, an ordered pair is a pair of numbers which are used to locate the point on a coordinate plane. Ordered pairs are also used to show the position on a graph where the horizontal value (x') is the first and vertical value (y') is second. If the ordered pairs have two elements, it is written in the form (x, y) in which x is fixed as the second component. For example,
We can pair off the elements and member diagrammatically as follows: -
Elements | Numbers |
x | 20 |
y | 30 |
z | 40 |
We have drawn an arrow from the elements to the numbers. Such figure is called a balloon diagram or arrow diagram. The arrows are used to show the relationship between the ordered pairs.
When the first component and the second component of an ordered pair are correspondingly equal then it is called equality of ordered pairs. For examples,
(x, y) = (x, y)
(-2, -5) = (-2, -5)
(1, 2) = (1, 2)
(4, 5) = (\(\frac{8}{2}\), \(\frac{10}{2}\))
Cartesian product is simply defined as the set of all possible ordered pairs with first element x and second element y.
Mathematically, the cartesian product of two sets X and Y is written as,
X × Y = {(x, y) ; x ∈ X and y ∈ Y}
The cartesian product can be represented in 3 ways. They are as follows:
Tree Diagram
The tree diagram can be represented as follows:
Let, suppose two sets X = (x, y, z) and Y = (3, 4).
Now, Taking x-component from set x and y-component from set y, then we can do all possible pairs as given below:
\(\therefore\) X × Y = {(x, 3), (x, 4), (y, 3), (y, 4), (z, 3), (z, 4)}
Similarly, we can find the cartesian product B × A as follows:
B × A = {(3, x) (3, y) (3, z) (4, x) (4, y) (4, z)}
Mapping Diagram
Mapping diagram can be represented as given below:
X= {3, 4} and Y = {5, 6}
X×Y = {3, 4} × {5,6} = {(3, 5) (3, 6) (4, 5) (4, 6)}
Here, Each arrow represented ordered pairs of A × B.
Graphical Representation
Graphical representation can be represented as below:
Suppose, X = {1, 2, 3} and Y = {2, 4, 6}
Then {X×Y} = {1, 2, 3} × {2, 4, 6} = {(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)}
Solution:
Here,
(4a, 6b) = (8, 12)
Comparing the corresponding first and second components,
or, 4a = 8
or, a = \(\frac{8}{4}\)
or, a = 2
And,
or,6b = 12
or, b = \(\frac{12}{6}\)
or, b = 2
∴ a = 2 and b = 2.
Solution:
Here,
(5m+1,−10) = (16, 3n+2)
Comparing the corresponding first and second cpmponents,
or, 5m + 1 = 16
or, 5m = 16 − 1
or, 5m = 15
or, m = \(\frac{15}{5}\)
or, m = 3
And,
or, 3n + 2 = −10
or, 3n = −10 − 2
or, 3n = −12
or, n = \(\frac{−12}{3}\)
or, n = −4
∴ m = 3 and n = −4.
Solution:
Comparing the coressponding first and second components,
Here,
(2x, 3y) = (−6, 9)
or, 2x = −6
or, x = \(\frac{−6}{2}\)
or, x = −3
And,
or, 3y = 9
or, y = \(\frac{9}{3}\)
or, y = 3
∴ x = −3 and y = 3.
Solution:
Here,
(n− 2, \(\frac{m}{4}\)) = (2, 0)
Comparing the corresponding first and second components,
or, n − 2 = 2
or, n = 2 + 2
or, n = 4
And,
or,\(\frac{m}{4}\) = 0
or, m = 0
∴ n = 4 and m = 0
Solution:
Here,
(35, −5) = (a − 2, b + 3)
Comparing the corresponding value First and second components,
or, a − 2 = 35
or, a = 35 + 2
or, a = 37
And,
or, b + 3 = −5
or, b = −5 − 3
or, b = −8
∴ a = 37 and b = −8
Solution:
Here,
(\(\frac{m}{2}\), 3) = (3, \(\frac{n}{2}\) + 1)
Comparing the corresponding first and second components,
or, \(\frac{m}{2}\) = 3
or, m = 3 × 2
or, m = 6
And,
or, \(\frac{n}{2}\) + 1 = 3
or, \(\frac{n}{2}\) = 3 − 1
or, \(\frac{n}{2}\) = 2
or, n = 2 × 2
or, n = 4
∴ m = 6 and n = 4
Solution:
Here,
(−x, 3) = ( 4, −y)
Comparing the corresponding first and second components,
or, −x = 4
or, x = −4
And,
or, −y = 3
or, y = −3
∴ x = −4 and y = −3
Solution:
Here,
(5x, 7y) = (15, 28)
Comparing thr corresponding first and second components,
or, 5x = 15
or, x = \(\frac{15}{5}\)
or, x = 3
And,
or, 7y = 28
or, y = \(\frac{28}{7}\)
or, y = 4
∴ x = 3 and y = 4
Solution:
Here,
(a − 2, b + 1) = (4, 2b)
Comparing the corresponding first and second components,
or,a − 2 = 4
or, a = 4 + 2
or, a = 6
And,
or, b + 1 = 2b
or, 2b − b = 1
or, b = 1
∴ a = 6 and b = 1
Solution:
Here,
(p + 4, 6) = (10, q)
Comparing the corresponding first and second components,
or, p + 4 = 10
or, p = 10 − 4
or, p = 6
And,
or, q = 6
∴ p = 6 and q = 6
Solution:
Here,
(a + 3, b) = (10, 5)
Comparing the corresponding first and second components,
or, a + 3 = 10
or, a = 10 − 3
or, a = 7
And,
or, b = 5
∴ a = 7 and b = 5
Solution:
Here,
(6,−y) = (−x,−4)
Comparing the corresponding first and second components,
or, −x = 6
or, x = −6
And,
or, −y = −4
or, y = 4
∴ x = −6 and y = 4
Solution:
Here,
(a, 3) = (4, b + 1)
Comparing the corresponding first and second components,
or, a = 4
And,
or, b + 1 = 3
or, b = 3 − 1
or, b = 2
∴ a = 4 and b = 2
Solution:;
Here,
(x, y − 2) = (6, 8)
Comparing the corresponding first and second components,
or, x = 6
And,
or, b − 2 = 8
or, b = 8 + 2
or, b = 10
∴ x = 6 and b = 10
Solution:
Here,
(−p, 6q) = (3, 36)
Comparing the corresponding first and second components,
or,−p = 3
or, p = −3
And,
or, 6q = 36
or, q = \(\frac{36}{6}\)
or, q = 6
∴ p = −3 and q = 6
Find the values of a and b.
(a-1, b)= (2, 3)
Find the values of a and b.
(a+3, b-4) = (0, 5)
Find the values of a and b.
(2a-3, 4) = (4a, b+5)
Given pairs are equal. Find x and y.
(6, y) and (x, 5)
For what value of x and y, (x+5, y+2) and (7, 5) are equal to each other.
For what value of a and b, (a+5, b+2) and (7, 5) are equal to each other.
For what value of a and b, (2a+5, b+2) and (7, 4) are equal to each other.
For what value of a and b, (3a-4, b-2) and (a+8, 2b-5) are equal to each other.
For what value of a and b, (2a+5,b+2 ) and (a+7, 2b-5) are equal to each other.
Find the value of and b.
(2a-1, b+2) = (1, 2)
Find the value of and b.
(a+b, a-b) = (8,0)
Find the value of and b.
(a+b, b+3) = (6, 2b)
Given pairs are equal. Find the value x and y.
(3x+2, 2y+1) and (5, 3)
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