What is Cartesian Product?
Cartesian Product is a multiplication of two sets in which we get a set of ordered pairs as a result.
For Example;
Consider two non empty sets A & B with following elements.
A = { a1, a2 }
B = { b1, b2, b3 }
The Cartesian Product of A X B is given as:
Note that the cartesian product is a collection of ordered pair.
There are total of 6 ordered pair in above multiplication.
In the ordered pair, the first element is from set A and second element is from set B.
Conclusion
Ordered pair is simply multiplication of two sets which results in a set of ordered pairs.
How to do cartesian product of two sets?
Suppose two sets A & B are given;
A = { 4, 5, 7 }
B = { 10, 12 }
For multiplication of A x B, follow the below steps:
(i) Form ordered pair of first element of set A with first and second element of set B.
(ii) Now form ordered pair of second element of set A with all elements of set B.
(iii) Finally form ordered pair of third element of set A with all elements of set B.
Combining all the ordered pair in one set we get;
A x B = {(4, 10), (4, 12), (5, 10), (5, 12), (7, 10), (7, 12) }
Number of Ordered Pair in Multiplication
Suppose there are two sets P & Q with following elements.
P = { a1, a2, a3 }
Q = { b1, b2 }
Here the set P contains three elements; a1, a2 & a3.
So n(P) = 3
Set Q contains two elements; b1 & b2 .
So n(Q) = 2
The number of ordered pair in multiplication of P x Q is given by following formula;
⟹ n(P) x n(Q)
⟹ 3 x 2
⟹ 6
Hence, we will get total of 6 ordered pairs after multiplication.
The exact ordered pairs are shown below;
Note the 6 ordered pairs which are formed after multiplication of sets P & Q.
Conclusion
The number of ordered pairs after multiplication of two sets can be calculated by multiplying the number of individual elements of set.
Is multiplication of A x B is same as B x A ?
No!!
The multiplication of A x B will yield completely different results that B x A.
Let us consider one example.
Given below is the set A & B with two elements each.
A = { a1, a2 }
B = { b1, b2 }
First Multiply A x B
Now Multiply B x A
Both the multiplication contain completely different ordered pairs.
Note that the ordered pair (a1, b1) is not equal to pair (b1, a1).
Hence, all the pairs in above multiplication is completely different.
Solved Problems – Cartesian Product of Set
(01) Given below are two sets A & B.
A = { 3, 5 }
B = { 4, 2 }
Find A x B and B x A
Solution
Finding A x B
A x B = { (3, 4), (3, 2), (5, 4), (5, 2) }
Finding B x A
B x A = { (4, 3), (4, 5), (2, 3), (2, 5) }
(02) Given below are two sets P & Q with following elements.
P = { 3, 9, 7, 1}
Q = { 5, 12, 4 }
Find the number of ordered pairs in P x Q
Solution
P = {3, 9, 7, 1 }
There are 4 elements in set P.
n(P) = 4
Q = { 5, 12, 4 }
There are 3 elements in set Q
n(Q) = 3
In P x Q, the number of ordered will be;
⟹ n(P) x n(Q)
⟹ 4 x 3
⟹ 12
Hence, in P x Q there will be 12 ordered pairs.
(03) Given below are two sets A & B with following elements
A = { 4, 6, 8 }
B = { 1, 7 }
Find the following products.
(i) A x A
(ii) A x B
(iii) B x A
Solution
(i) A x A = { 4, 6, 8 } x { 4, 6, 8 }
A x A = { (4, 4), (4, 6), (4, 8),(6, 4), (6, 6), (6, 8),(8, 4), (8, 6), (8, 8) }
(ii) A x B = { 4, 6, 8 } x { 1, 7 }
A x B = { (4, 1), (4, 7), (6, 1), (6, 7), (8, 1), (8, 7) }
(iii) B x A = { 1, 7 } x { 4, 6, 8 }
B x A = { (1, 4), (1, 6), (1, 8), (7, 4), (7, 6), (7, 8) }
(04) Given below are two sets A & B.
A = { 9, 2, 4 }
B = { 𝜙 }
Find A x B.
Solution
Notice that set B is a null set which means that it doesn’t have any elements.
The multiplication of any set with null set results in null set.
It’s just similar to multiplication of number with 0 results in 0.
A x B = { 9, 2, 4 } x { 𝜙 } = 𝜙
A x B = { 𝜙 }