Ordered Pairs and Cartesian Product

Transcript

0:00Introduction

Hello everyone! Let's start with one quick question. Is the set one, two the same as the set two, one? Yes, exactly! In a set, order does not matter. Now, on a graph, is the point one comma two the same as the point two comma one? No, no! Two different points. Same numbers, different order, different thing. That is today's topic: ordered pairs, and the Cartesian product. And JEE loves it: counting pairs, counting subsets of pairs, and one sneaky trap with unions. So, let's begin.

0:36Ordered pair: order matters

So, what is an ordered pair? Two things in round brackets, where order matters: a first entry, and a second entry. On screen: a comma b equals c comma d, if and only if a equals c, and b equals d, both together, okay? The double arrow is read, if and only if. Now our puzzle. In one comma two and two comma one, the first entries are one and two. Not equal! So these pairs are different. The slashed equals sign means, not equal. But the sets in curly brackets are equal, because in a set, order does not matter. Clear? One small warning. Round brackets with two entries can also mean an interval on the number line, so read the context. And an ordered pair is never the set in curly brackets. Different brackets, different meaning, okay?

1:30Cartesian product: every possible pair

Now the big idea. The Cartesian product of two sets, written A cross B, with the little cross sign on screen, is the set of all ordered pairs whose first entry comes from A, and second entry comes from B, okay? The colon is read, such that, and the curved symbol means, belongs to. Every possible pair, nothing missing. Look at the picture. Here, set A is one, two, three, along the bottom, and set B is p, q, up the side. Every dot is one ordered pair: one comma p, two comma p, three comma p, and the same three with q. Just like seats in a hall: column, then row. Six dots. But tell me, why exactly six? Yes! Three choices for the first entry, and for each of them, two choices for the second: three times two, six. So, n of A cross B equals n of A, times n of B, okay? The dot on screen means times. Times, not plus! Students in a hurry add. Don't! Same idea with three sets. Set A cross B cross C is the set of ordered triplets: first entry from A, second from B, third from C, okay? And the count? Exactly the same pattern: n of A, times n of B, times n of C, okay? And you have already met this. The bold R stands for the real numbers. Real numbers cross real numbers, R with a little two on top, is the coordinate plane. And R with a little three on top is space. Looks familiar, right?

3:06Properties you must know

Now some properties. These are favourites in JEE options. First, the empty set, the circle with a slash. If one set is empty, there is nothing to choose for that entry, so there are no pairs at all. So, A cross the empty set is empty, and the other way round too. Now tell me. When is A cross B equal to B cross A? Hmm. Order matters, right? So usually they are different. Yes, only in two cases: when the two sets are equal, or when one of them is empty. Students forget the second case. Don't! Next, cross works nicely with union, the cup sign. So, A cross, B union C, equals A cross B, union, A cross C, okay? Just like opening a bracket in algebra. And the same works for intersection, the upside down cup, and for difference, the minus sign. For example, A cross, B minus C, equals A cross B, minus, A cross C, okay? That's it: three rules, one pattern. Got it? One more, very useful. The intersection of two products: A cross B, intersection, C cross D, equals A intersection C, cross, B intersection D, okay? First entries come from both A and C, second entries from both B and D, clear?

4:32JEE trap: union does not split this way

Alright, now the JEE trap. Intersection of two products splits nicely. So, A cross B, union, C cross D, must be A union C, cross, B union D, right? Same pattern, looks perfect. No, no, careful! For union, this is not true in general. See the slashed equals sign. Don't believe me? Let's test it. Topper's habit: whenever you doubt a rule, test it on tiny sets. Take sets A and B both equal to the set with just one, and sets C and D both equal to the set with just two. Left hand side first. Set A cross B gives only one comma one, and C cross D gives only two comma two. So the union has just these two pairs. Right hand side. Set A union C is one, two, and B union D is also one, two. Cross them, and we get all four pairs, including one comma two, and two comma one. Extra pairs! So the sides are not equal. Got it? The intersection of two products splits, but the union of two products does not.

5:41Example 1: is A x B equal to B x A

Let's do example one: our opening puzzle, now with sets. Set A is one, two, and set B has three letters: small a, small b, small c. Write A cross B and B cross A, and tell me, are they equal? Pause the video, and try it first. Done? Okay, let's check. In A cross B, the first entry comes from A, okay? So pair one with each letter, then two with each letter. Six pairs. In B cross A, the letters come first: small a comma one, small b comma one, and so on. Again, six pairs. Two times three, and three times two: six both times. The same count, no doubt. So, same size means same set, right? No! Look at one comma small a, and small a comma one. Same two things, different order, so different pairs. So, A cross B is not equal to B cross A, exactly like our opening puzzle. Same size does not mean same set, okay?

6:51Example 2: subsets of A x B with at least 3 elements

Now a real JEE question, from JEE Main twenty fifteen. Set A has four elements, and set B has two. How many subsets of A cross B have at least three elements? Pause and try. Hint: count what you don't want, and subtract. Done? Okay, let's check. How many pairs in A cross B? Four times two, eight. Not four plus two, okay? So we have a set with eight elements. It has two to the power eight subsets: two hundred and fifty six. Counting three, four, five, up to eight elements directly is long. So, we remove the small ones. Which subsets have fewer than three elements? One element: eight of them. Two elements: eight choose two, twenty eight. So the answer is two hundred and fifty six minus thirty six, two hundred and twenty, right? Oh no, careful! We forgot the empty set, with zero elements! It also has fewer than three. The tall brackets on screen are read eight choose zero, eight choose one, and eight choose two. One plus eight plus twenty eight: thirty seven to remove. So the answer is two hundred and fifty six minus thirty seven: two hundred and nineteen. Look, just two ideas: count the pairs by multiplying, and never forget the empty set.

8:20Quick revision: ordered pairs, n(A x B), the union trap

So, quick revision. One: in an ordered pair, order matters, so one comma two is not two comma one. Two: n of A cross B is n of A times n of B, okay? Multiply, never add. Three: cross splits over union, intersection and difference, but a union of two products does not. When in doubt, test tiny sets. Now go and try the practice problems on this page. You will enjoy them, I promise. All the best!