To solve this question, let's break it down into simpler parts and then use the given sets to find the correct answer.
We are given the expression [tex]\( X \times Y = \{(a, b) ; a \in X, b \in Y\} \)[/tex], where [tex]\( n(X) = n \)[/tex] and [tex]\( n(Y) = m \)[/tex]. Therefore, the number of elements in the Cartesian product [tex]\( X \times Y \)[/tex] is [tex]\( n(X \times Y) \)[/tex].
Step 1: Understanding Cartesian Product
- The Cartesian product [tex]\( X \times Y \)[/tex] consists of all possible pairs [tex]\( (a, b) \)[/tex] where [tex]\( a \)[/tex] is from set [tex]\( X \)[/tex] and [tex]\( b \)[/tex] is from set [tex]\( Y \)[/tex].
Step 2: Calculating Number of Elements
- For any two sets [tex]\( X \)[/tex] and [tex]\( Y \)[/tex], if [tex]\( X \)[/tex] has [tex]\( n \)[/tex] elements and [tex]\( Y \)[/tex] has [tex]\( m \)[/tex] elements, then the Cartesian product [tex]\( X \times Y \)[/tex] will have [tex]\( n \times m \)[/tex] elements.
- Therefore, the number of elements in [tex]\( X \times Y \)[/tex] is [tex]\( n(X \times Y) = n \times m \)[/tex].
Step 3: Applying This to Given Sets
- Given sets are [tex]\( A = \{1, 2\} \)[/tex] and [tex]\( B = \{1, 2, 3, 4\} \)[/tex].
- Number of elements in set [tex]\( A \)[/tex] is [tex]\( n(A) = 2 \)[/tex].
- Number of elements in set [tex]\( B \)[/tex] is [tex]\( n(B) = 4 \)[/tex].
Step 4: Calculating [tex]\( n(A \times B) \)[/tex]
- Based on our earlier calculation, the number of elements in [tex]\( A \times B \)[/tex] is [tex]\( n(A) \times n(B) = 2 \times 4 = 8 \)[/tex].
Step 5: Identifying the Correct Answer
- Given options:
- (1) [tex]\( m^2 n^2 \)[/tex]
- (2) [tex]\( m n \)[/tex]
- (3) [tex]\( m^2 n^3 \)[/tex]
- (4) [tex]\( m^3 n^2 \)[/tex]
- From our calculation, we see that [tex]\( n(A \times B) = 8 \)[/tex], which aligns with the expression [tex]\( m \times n \)[/tex].
Therefore, the correct answer is option (2) [tex]\( m n \)[/tex].
