Inverse of a Relation

Transcript

0:00Introduction

Hello everyone! Here is one small puzzle before we start. Take the relation, less than, on the real numbers: x is related to y when x is less than y. Now I do one simple thing. I flip every pair: the first entry becomes the second, and the second becomes the first. Tell me, what relation do I get now? Still less than? Or something new? Hmm, keep thinking. By the end of this class, you will answer it in seconds. Today's topic is the inverse of a relation. And look, JEE uses this flipping idea again and again. So, let's begin.

0:37Inverse: reverse every pair

So, start with a relation R, from a set A to a set B, okay? Remember, R is just a set of ordered pairs. In every pair, the first entry comes from A, and the second entry comes from B, clear? Now the inverse. We write it as R with a little minus one on top, and we read it, R inverse. R inverse is the set of all pairs b comma a, such that a comma b belongs to R, okay? The colon is read, such that, and the curved symbol means, belongs to. In simple words? Take every pair of R, and turn it around. That's it! Got it? Now tell me. R goes from A to B, and in every pair, the first entry is from A, okay? After flipping, where does R inverse go? From B to A, exactly! Now the first entries come from B, and the second entries from A, okay? So R inverse is a relation from B to A, just the other way round. And if you draw R as an arrow diagram, with arrows going from A to B? Then for R inverse, just reverse every arrow. Same arrows, opposite direction. Really simple.

1:43Domain and range swap places

Alright, now the most useful property. The domain of R is the set of all first entries, and the range of R is the set of all second entries, okay? Now we flip every pair. So the domain of R inverse is just the domain of R, right? Hmm. No, no, careful! This is exactly where students lose marks. After flipping, the old second entries are now in front, okay? On screen, Dom is short for domain. So, the domain of R inverse equals the range of R, the old second entries. And the same way, the old first entries are now at the back. So the range of R inverse equals the domain of R, okay? See, domain and range simply swap places. Once you know them for R, you never calculate them again! One more question. When we flip, can we gain a new pair, or lose a pair? Neither, that's right! Every pair of R gives exactly one pair of R inverse, and two different pairs stay different after flipping. So n of R inverse equals n of R, okay? The same number of pairs.

2:51Flip twice, and flip a union

Now, some quick properties. Suppose you flip every pair of R, and then you flip them all once again. What do you get? Back where you started, yes! So the inverse of R inverse is R itself, okay? Flip a coin over twice, and you see the same face again. Next, take two relations: R, and another one, S, okay? The inverse of R union S, the cup symbol, equals R inverse, union, S inverse. Why? Flipping a pair does not care which relation the pair came from. Just flip everything. And it is exactly the same with intersection, the upside down cup. The inverse of R intersection S equals R inverse, intersection, S inverse. Two rules, one pattern. Clear?

3:38Topper's trick: the rule for R inverse

Now a trick that toppers use. Often, JEE does not give R as a list. It gives a rule in x and y. So how do you write R inverse as a rule? Look at the definition again. The pair x comma y is in R inverse if and only if the pair y comma x is in R, okay? The double arrow on screen is read, if and only if. So, just swap x and y in the rule of R, and you have the rule for R inverse. No need to flip pair by pair. We will use this in our very first example. With me?

4:12Example 1: R inverse of x + 2y = 8, domain and range

Let's do example one. Here, R has three pairs: six comma one, four comma two, and two comma three. They come from the rule, x plus two y equals eight, on the natural numbers, the bold N on screen. Find R inverse, with its domain and range. Pause the video, and try it first. Done? Okay, let's check. Flip every pair. Six comma one becomes one comma six, four comma two becomes two comma four, and two comma three becomes three comma two. Three pairs in, three pairs out, as promised. Now the rule, with our topper's trick. Swap x and y in x plus two y equals eight, and you get two x plus y equals eight, with x and y still natural numbers, okay? Quick check with one comma six: two times one, plus six, is eight. Spot on. Now the domain and range of R inverse. Do we need to calculate anything new? Not at all! Domain and range just swap. The range of R is one, two, three, so the domain of R inverse is one, two, three. And the domain of R is six, four, two, so the range of R inverse is two, four, six. Same set, just written in increasing order. Got it?

5:31Example 2: inverse of the less than relation

Now a harder one, and it is our opening puzzle. R is the relation, x less than y, on the real numbers. The bold R in the question means the real numbers, and R cross R means all pairs of real numbers, okay? Describe R inverse, then R intersection R inverse, and then R union R inverse. Pause the video, and try it. Hint: flip, then rename the letters. Done? Okay, let's check. Flip every pair: R inverse is the set of pairs y comma x, where x is less than y. Now rename the letters, so the first entry is called x again. Then R inverse is the set of pairs x comma y with x greater than y. So the inverse of less than is greater than! There is the answer to our puzzle. Solved in seconds, right? Now R intersection R inverse: pairs where x is less than y, and also x is greater than y. Less than and greater than, together. So x equals y, the line y equals x, right? No, no, careful! This is the JEE trap. Both are strict: less than, and greater than. Can x be less than y and greater than y at the same time? Never! And on the line y equals x, x is not less than y, so those points are in neither relation. So the intersection is empty, the circle with a slash on screen. Now the union: pairs where x is less than y, or x is greater than y. That covers every pair, except the pairs where x equals y. So R union R inverse is the set of pairs with x not equal to y. The slashed equals sign means, not equal. And in the plane? Everything, except the line y equals x. Look, one flip, and we got all three answers. As simple as that.

7:26Quick revision: reverse pairs, swap domain and range

So, quick revision. One: to get R inverse, reverse every pair, or reverse every arrow. Two: the domain and range swap places, and the number of pairs stays the same. Three: for a rule, just swap x and y in the rule of R, okay? Now go and try the practice problems on this page. You will find them easy, I promise. All the best!