Question

What is the Cartesian product of \[A=\{1,2\}\] and \[B=\{a,b\}\]?
(A) \[\{(1,a),(1,b),(2,a),(b,b)\}\]
(B) \[\{(1,1),(2,2),(a,a),(b,b)\}\]
(C) \[\{(1,a),(2,a),(1,b),(2,b)\}\]
(D) \[\{(1,1),(a,a),(2,a),(1,b)\}\]

Answer 1

Hint: To obtain the Cartesian product of two sets we should multiply every element of set A with every element of set B. Both the sets should be non-empty sets. To obtain the Cartesian product both the sets should be multiplied in a specific order otherwise the Cartesian product will be wrong.

Complete step-by-step answer:
A collection of ordered pairs of elements is known as the set. A set B is the subset of another set A if every element of set B is in set A.
If there are two sets A and B, then their Cartesian product will be obtained as follows. The first element of set A will be multiplied with both the elements of set B and then the second element of set A will be multiplied by both the elements of set B. So, the Cartesian product of two sets means the product of two non-empty sets and it is also the collection of all ordered pairs of elements. The ordered pair is the set of two elements in a specific order. If two sets are null then their Cartesian product will also be a null set.
If there is an ordered pair \[(a,b)\] then a is known as the first component and b is known as the second component in the ordered pair and the two ordered pairs will be equal if both of their first and second components are equal to each other.
If the number of ordered pairs in set A is \[n(A)\] and the number of ordered pairs in set B is \[n(B)\] then the number of ordered pairs in their Cartesian product will be \[n(A\times B)\]. The cartesian product does not obey the commutative property. Also, the Cartesian product does not obey the associative property.
The Cartesian product of \[A=\{1,2\}\] and \[B=\{a,b\}\] will be given as shown below.
\[A\times B=\{(1,a),(1,b),(2,a),(2,b)\}\]
The correct answer will be an option(C) \[\{(1,a),(1,b),(2,a),(2,b)\}\].
So, the correct answer is “Option C”.

Note: A set that does not contain any element is known as an empty set or null set. The set that contains exactly one element is known as a singleton set. A set that contains at least one element is known as a non-empty set. The intersection of two sets represents the elements which are common in both sets.