A pair of numbers which always follows a rule that the first components should always be X-component and the second component should always be taken from Y-component, where the pair must be always enclosed within small brackets () and separated by a common is known as an ordered pair.
An ordered pair is always in the form of (x, y). The first component, i.e x-component is also known as the antecedent and the second component, i.e y-component is also called the consequent.
Two ordered pairs are said to be equal if and only if their corresponding components are equal.
i.e. (a,b) = (c,d) if a = c and b = d
The Cartesian Product between two sets is the set of all possible ordered pairs with the first element from the first set and second element from the second set. Let A and B be two empty sets. Then the Cartesian product A×B (cartesian product of A and B) read as A cross B is defined as the set of all possible ordered pairs (x, y) such that x∈A and y∈B. Mathematically,
A×B = {(x, y)x∈A and y∈B}
For example,
Let A = {1,2}, B = {2,1}. Prove thatA×B = B×A.
Here, A= {1,2}
B = {2,1}
∴A×B = {(1,2), (1,1), (2,2), (2,1)}
B×A = {(2,1)(2,2)(1,1)(1,2)}
Since all the ordered pairs in A×B and B×A are the same, A×B = B×A.
The cartesian product can be represented in various ways:
Soln
Here,
given (4,y) and (x,7) are equal,so
4=x and y=7
∴ x=4 and y=7.Ans
Soln,
Here given (x+y,2)=(1,x-y)
Equating the corresponding terms
x+y=1 ..........(i) and 2=x-y ...........(ii)
adding equations (i) and (ii), we get,
x+y=1
\(\frac{x-y=2}{2x=3}\)
∴ x=\(\frac{3}{2}\)
putting x=\(\frac{3}{2}\) is (i), we get,
\(\frac{3}{2}\)+y=1
or, y=1-\(\frac{3}{2}\)=\(\frac{2-3}{2}\)=\(\frac{-1}{2}\)
∴ x=\(\frac{3}{2}\) and y=\(\frac{-1}{2}\) Ans.
Soln
Since the ordered pair is equal.
2x = 4
or, x = \(\frac{4}{2}\) = 2
Again, \(\frac{y}{3}\) = 3
or, y = 9
\(\therefore\) x = 2 and y = 9.
Soln
Since the ordered pairs are equal,
3x-5 = 10
or, 3x = 10 + 5
or, x = \(\frac{15}{3}\) = 5
Gain, 2y + 2 = 3y - 1
or, 2 + 1 = 3y - 2y
or, 3 = y
\(\therefore\) x = 5 and y = 3.
Since the ordered pairs are equal,
x+y = 5........(i)
x - y = 3.....(ii)
Adding equation (i) and (ii), we get,
2x = 8
or, x = 4
Replacing the value of x in equation (i),
4 + y = 5
or, y = 5 - 4 = 1
\(\therefore\) x = 4 and y = 1
Soln
Here the ordered pairs (a,9 and (1,b) belongs to the set {(x,y): y = 2x + 3}
So, If (a,9) {(x,y): y = 2x + 3}
a = x and 9 = y
\(\therefore\) 9 = 2.a + 3
or, 2a = 6
\(\therefore\) a = \(\frac{6}{2}\) = 3
Similarly, If (1,b) {(x,y): y = 2x + 3}
1 = x and b = y
\(\therefore\) b = 2.1 + 3
= 2 + 3 = 5
hence, a = 3 and b = 5.
If in a relation R={(x,y ):x-2y=5} has the ordered pairs(7,a) nd (9,b),Find the values of a and b.
a =1,b=2
a =1,b=1
a =1,b=3
a =2,b=2
Find the values of x and y if (x, -2) = (4, y) in equal ordered pairs.
Find the values of x and y if (x + y, y + 3) = ( 6, 2y) equal ordered pairs.
If the ordered pairs (a, 9) and (1, b) belongs to the set {(x, y) : y = 2x + 3}, find the values of a and b.
Find the values of the x and y if (x + y, x - y) = (8, 0) in equal ordered pairs.
Find the values of x and y if (2x - 1, y + 2) = (-1, 2) in equal ordeed pairs.
Find the values of x and y if (2^{x}, 3^{y}) = (16, 27) in equal ordered pairs.
If A = {a, b}, B = {c, d} and C = {d, e}, find A × (B∩C).
If A = {a, b}, B = {c, d} and C = {d, e} find (A∩B) × (B∩C}.
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