Direct product

Imagine you have two sets of toys, one set has red cars and blue balls, and the other set has yellow trucks and green blocks. If you wanted to combine these two sets, you could create a new set by taking one toy from the first set and one toy from the second set to make a pair. For example, you could take a red car from the first set and a yellow truck from the second set to make a new pair.

This idea of combining two sets to create a new set is very similar to what is called a direct product in mathematics. When we talk about a direct product of two sets, we are referring to a new set that combines the elements of each original set to create pairs. Each pair is made up of one element from the first set and one element from the second set.

To use our toy example again, the direct product of the set of red cars and blue balls with the set of yellow trucks and green blocks would be a new set containing pairs such as (red car, yellow truck) and (blue ball, green block).

In more math-y terms, if we have two sets A and B, the direct product of A and B (denoted as A × B) is a new set made up of all possible ordered pairs (a,b), where a is an element of A and b is an element of B.

Overall, the idea of a direct product is just a way to combine two sets into a new set of pairs.