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Problem 4

# If $$A$$ has $$a$$ elements and $$B$$ has $$b$$ elements, how many elements are in $A \times B$ ? Explain your answer.

Expert verified
The number of elements in the Cartesian product $$A \times B$$ is equal to $$a \times b$$, where $$a$$ is the number of elements in A and $$b$$ is the number of elements in B. This is because each element in A can be paired with each element in B, resulting in a total of $$a \times b$$ ordered pairs.
See the step by step solution

## Step 1: Understanding Cartesian Product

A Cartesian product of two sets A and B, denoted as $$A \times B$$, is the set of all possible ordered pairs where the first element is from A and the second element is from B. The Cartesian product of two sets results in a new set containing all possible combinations of elements from the two sets.

## Step 2: Counting Elements in A

Set A has $$a$$ elements. For each element in A, there will be an ordered pair formed with the elements of B.

## Step 3: Counting Elements in B

Set B has $$b$$ elements. For each element in B, there will be an ordered pair formed with the elements of A.

## Step 4: Counting Elements in the Cartesian Product

Since there are $$a$$ elements in A and $$b$$ elements in B, there will be $$a \times b$$ combinations for the ordered pairs. Hence the number of elements in the Cartesian product $$A \times B$$ is equal to $$a \times b$$.

## Step 5: Result

The Cartesian product $$A \times B$$ has $$a \times b$$ elements, where $$a$$ is the number of elements in A and $$b$$ is the number of elements in B.

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