the use of Cartesian product I am currently studying in secondery level. I read cartesian product the other day and I found it absolutely bizarre. In my text book, there is this "order pair" which I understood fairly well and then there is cartesian product in which we multiply two sets. I don't understand the concept behind it. What is its application? How will it come to use in higher level study?
The basic point of set theory in mathematics is that it allows us to wrap up properties and treat them as if they were things.
Without sets we can say things like

$2.5$ is a rational number.
$\pi$ is not a rational number.
$36$ is a perfect square.
$(0,42)$ is an ordered pair of two whole numbers.

The wordings &quot;is a rational number&quot;, &quot;is a perfect square&quot;, &quot;is an ordered pair of two whole numbers&quot; denote properties. They are nice for talking about particular things that have or don't have the property. But our language doesn't allow us to speak about, for example, an expression whose value is a property, or a function that takes a property as input or things like that.
This is where sets come in. A set is a way to &quot;box up&quot; a property such that it becomes a &quot;thing&quot; that we can speak about together with other similar things. To a large extent, a property and a set are different ways of thinking about the same concept. If we have a property, say &quot;is a thingamajig&quot; we can always (with some technical exceptions that you don't need to worry about at this level) create a set from it: $$A=\{x\mid x\text{ is a thingamajig}\}$$
On the other hand, whenever we have a name or an expression for a set, this gives us a property: &quot;is a member of such-and-such set&quot;.
Because sets are &quot;things&quot;, they can have (or not have) properties themselves -- such as the property &quot;is a set of real numbers (but possibly not all the real numbers)&quot;. Boxing up these properties, we get sets of sets and so forth.
Now the cartesian product $A\times B$ is neither more nor less than the set corresponding to the property &quot;is an ordered pair of something from $A$ and something from $B$&quot;.
Thus, whenever we want to speak about ordered pairs at all, it may suddenly become useful to speak about the property of being an ordered pair. And then it immediately becomes useful to speak about the boxed-up property that corresponds to it. In fact so useful that the common way to say &quot;is an ordered pair of something from $A$ and something from $B$&quot; is &quot;is in $A\times B$&quot;. This is shorter and more concise.
So cartesian products are used, or potentially used, everywhere we use ordered pairs in the first place.
also i just got to learn that we use set cartesian products in combinatorial problems .An example of this is a three course meal where there is an appetiser main course and desert.Lets say for example we have porridge,soup,lemonade and main course chicken,omlades,cooked potatoes,and desert icecream fruitpoodingand chocolate one can have soup,chicken,chocolate or soup,omlates,fruitpooding etc thus such is an example of application of set cartesian product