If A = {1, 5} and B = {2, 4}, what is 𝐴×𝐵?{(1,1), (1,4), (5,2), (4,4)}{(1,2), (1,4), (5,2), (5,4)}{(1,1), (2,2), (5,5), (4,4)}None of the above

Break Down the Problem

1. We need to find the Cartesian product $A \times B$ of the sets $A$ and $B$.
2. The Cartesian product is defined as the set of all ordered pairs $(a, b)$ where $a$ is from set $A$ and $b$ is from set $B$.

Relevant Concepts

• The Cartesian product of two sets $A$ and $B$ is expressed mathematically as: $A \times B = \{ (a, b) \mid a \in A, b \in B \}$

Analysis and Detail

1. From set $A = \{ 1, 5 \}$, the elements are $1$ and $5$.
2. From set $B = \{ 2, 4 \}$, the elements are $2$ and $4$.
3. We will create all possible pairs:
   • Pairing $1$ with each element in $B$:
     • $(1, 2)$
     • $(1, 4)$
   • Pairing $5$ with each element in $B$:
     • $(5, 2)$
     • $(5, 4)$
4. Hence, the complete Cartesian product $A \times B$ is: $A \times B = \{ (1, 2), (1, 4), (5, 2), (5, 4) \}$

Verify and Summarize

• We have calculated all the pairs and confirmed each one has been generated correctly based on the definitions of the Cartesian product.
• Thus, the calculated sets correspond with the characteristics of their parent sets.

Final Answer

The Cartesian product $A \times B$ is $\{ (1, 2), (1, 4), (5, 2), (5, 4) \}$. Therefore, the correct option is:

• {(1,2), (1,4), (5,2), (5,4)}