Operations on Sets

Transcript

0:00Introduction

Hello everyone! So, one small question first. At a wedding, someone asks you: tea or coffee? And you say, both! People will give you a strange look, right? But in maths, your answer is perfectly correct, because in maths, or always includes both. Today: union, intersection, difference, complement and symmetric difference, the five operations on sets. Look, sets questions in JEE are built from these five, so let's make them crystal clear.

0:36Union and intersection

So, here is our picture. The big box is the universal set U, and inside it are two sets, A and B, okay? Every operation today is just a different region of this one picture. Now, let's shade them one by one. First, union, the symbol like a cup. So, A union B is the set of all x such that x belongs to A or x belongs to B, okay? This curved E means belongs to. Now tell me, are the middle elements, the ones in both sets, in the union? Yes, of course! Or includes both, remember? Tea and coffee, both allowed. Next, intersection: the same cup, upside down. Here the word is and. So, A intersection B is the set of all x which belong to A and also belong to B, okay? Only the middle part, where the circles overlap. Clear?

1:33Difference and complement

Alright, now difference. So, A minus B is the set of all x which belong to A, and do not belong to B, okay? That is the curved E with a slash. Start with A, and throw away everything that B also has. Only the left part survives. Now tell me. Is A minus B the same as B minus A? Look at the screen. B minus A means in B, but not in A: the right part. Completely different, right? So write this down: A minus B is not equal to B minus A, in general. Order matters. In fact, difference is neither commutative nor associative. Now the complement. So, A prime, with the little dash, is U minus A: all x in U which are outside A, okay? Look at the shading: the right part of B, and the whole box outside the circles too. Books also write a small c up top, like a power, or a bar over it. Exactly the same thing.

2:35Symmetric difference: exactly one set

Fifth operation: symmetric difference, the little triangle. So, A triangle B equals A minus B, union B minus A: the left part plus the right part. Which region is left out? Yes, the middle! Elements in exactly one of the two sets. Second way to write it: take the union and remove the intersection, so A union B, minus A intersection B, okay? Same region, see? In word problems, exactly one means symmetric difference.

3:08Useful rewritings

Here are all the pictures in one place. Look at the top right one, for A minus B: in A, and not in B, okay? But not in B means in B prime! So, A minus B equals A intersection B prime. Topper's trick: whenever you see a minus between sets, rewrite it as an intersection with a complement. Now suppose A is a subset of B, this rounded symbol, so A sits completely inside B, okay? Tell me, what is A union B? Just B, yes! And A intersection B? Just A, exactly. The double arrow means if and only if: all these statements say the same thing. The chain goes on: A minus B is the empty set, the circle with a slash, since nothing in A is outside B, okay? And B prime is a subset of A prime: anything outside B is surely outside A too. Got it?

4:07Two classic traps

Now, two traps. First: take the set one, two, and the set two, three. Can we find the first minus the second? Many students say no, because the second set is not inside the first! No, no, of course we can. Keep the elements of the first set not in the second. One stays, two goes. Answer: the set containing just the number one. So, A minus B does not need B to be a subset of A, okay? Second trap: the complement depends on the universal set. Take A as the natural numbers, the double struck N, okay? What is A prime? Honest answer: it depends on U! If U is the integers, the double struck Z, then A prime is zero, minus one, minus two, and so on. But if U is all real numbers, the double struck R, then A prime is much bigger: every real number that is not natural. Same A, different answers. Always check U first, alright?

5:08Example 1: union, intersection, difference, complement

Now, Example one. U is the numbers one to ten. Set A is one, two, three, four, five, and set B is four, five, six, seven. Find all seven. Pause the video, and try it first. Done? Okay, let's check. First, fill the Venn diagram. Four and five are in both, so they go in the middle. One, two, three are only in set A, and six, seven only in set B, okay? Eight, nine, ten are in neither: outside both circles, in the box. Union: everything in either circle, one to seven. Intersection: only the middle, four and five. Four and five are in both sets, and still in the union. Or includes both! Next, A minus B: the left part, one, two, three. And B minus A: the right part, six and seven. Different, aren't they? Now, A prime: everything in U outside A, okay? Inside the circles, that is six and seven. So, six, seven, right? No, no, careful! This is the trap. What about eight, nine and ten, outside both circles? They are in U, and not in A, okay? So A prime is six, seven, eight, nine, ten. Never forget the box! Symmetric difference: exactly one set, the left part and the right part. One, two, three, six, seven. That's it, the middle is gone. And the complement of A union B: outside both circles. Eight, nine, ten. All seven answers, from one diagram!

7:00Example 2: set operations on intervals

Now a harder one, with intervals. Set A is from minus two to five, square bracket at minus two, round bracket at five. Set B is from one to seven, round bracket at one, square bracket at seven. Square bracket means the end point is included, round means it is not, okay? Find all five. Pause and try. Hint: draw a number line. Done? Okay, let's check. Picture the number line: A and B together cover minus two to seven, with no gap. So the union is minus two to seven, both ends included. The overlap is one to five, but one is missing from B, and five is missing from A, okay? So the intersection is open at both ends. Now, A minus B: the part of A outside B, okay? From minus two up to one. But is the end point one in it, or not? Many students put a round bracket, since B starts at one. Hmm, think again. Look, one is in A, and one is not in B, because of that round bracket. So one stays! So, A minus B is minus two to one, closed at both ends. Same logic for B minus A: five to seven. Five is not in A, round bracket, right? So five is included. Last, A prime, with U as all real numbers. From minus infinity, the sideways eight, up to minus two, and from five to infinity. Minus two is in A, so A prime leaves it out: round bracket. Five is not in A, so A prime takes it: square bracket. At each end point, the bracket flips. Simple as that.

8:47Quick revision: union, intersection, difference, complement

So, quick revision. Union: in either set, both allowed. Intersection: in both. Difference, A minus B: in A, not in B, and order matters. Complement depends on U, and symmetric difference means exactly one. Now go and try the practice problems on this page. You will find them easy. All the best!