Universal Set and Venn Diagrams

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. I give you two sets. The first one: all the prime numbers, two, three, five, seven, and so on. The second one: all the negative integers, minus one, minus two, minus three, and so on. Now I want one big set, one big box, that holds both of them. Which box will you choose? Most students say natural numbers, straight away. Hmm. Is that right? Wait, hold that thought! By the end of this video, you will know the answer, and why so many students get it wrong. Today we learn this big box, called the universal set, and its picture, called a Venn diagram. And look, in JEE, Venn diagrams are the fastest tool for checking set identities and solving counting problems. So, let's begin.

0:56The universal set: the big box

So, what is this big box? In a particular discussion, the universal set, written with a capital U, is a set that contains all the sets under consideration as subsets, okay? Every set we are talking about must sit inside it. That is why we call it universal: for that discussion, it is the whole world. Clear? Here is an example. While studying number systems, we may take U equal to the real numbers: the capital R with a double line that you see on screen. Every set of numbers we use there sits inside the real numbers, okay? Now tell me. Is the universal set always the real numbers? No! Think of a chapter on triangles. Are triangles numbers? No, right? So the real numbers cannot hold them. So, while studying triangles, the universal set may be the set of all triangles. Every set of triangles in that discussion sits inside this one, okay? Alright, so see the pattern. The definition begins with the words, in a particular discussion. The universal set is not one fixed set for all of mathematics; you pick it to fit the discussion. Change the discussion, and the universal set can change too. Remember this one line. It will save you in both examples today. Got it?

2:18Venn diagrams: sets as pictures

Now, how do we draw all this? With a Venn diagram, named after John Venn. First, a rectangle. This whole rectangle, with the little capital U in its corner, is the universal set, okay? Everything in our discussion lives somewhere inside it, even the two circles you see. We will come to those circles in a second. Then, inside the rectangle, we draw circles, or any closed curves, for the subsets. Here is set A, and here is set B, okay? Now think. Why must the circles sit inside the rectangle, and never outside it? Yes, exactly! Because the universal set contains every set of the discussion as a subset. Nothing is allowed to stick out of the box. Now, the middle part, where the two circles overlap. Suppose some element belongs to both sets. Where will you put it? Correct! In the overlap, because it must be inside circle A and inside circle B at the same time. So the overlapping region shows the common elements. Clear? And if two sets have no common elements at all? Then there is nothing to put in the middle, so we simply draw the circles apart, like this. Keep this picture in mind. It comes back in our second example.

3:35Topper's habit: draw it first

Now, why do toppers love Venn diagrams so much? Tell me honestly. Which is easier to hold in your head: a long sentence full of symbols, or one simple picture? Yes, the picture, of course! That is exactly what a Venn diagram does: it turns an abstract statement into a picture. One quick drawing, and you can see the whole question at once, right? That is why the Venn diagram is the fastest tool for checking set identities. Soon, when we study the laws of sets, this same picture will be your best friend. With me? And it is the fastest tool for solving counting problems too, the kind JEE loves. So here is the topper's habit: whenever a sets question looks confusing, draw the rectangle and the circles first, then think. Really simple, but it works.

4:25Example 1: choose a universal set for three sets

Let's try one. Set A is two, four, six. Set B is one, three, five. And set C has just one element, seven. Suggest a universal set for these three sets. Pause the video, and try it first. Done? Okay, let's check. First, collect every element that appears. From set A: two, four, six. From set B: one, three, five. And from set C: seven. So, all the numbers from one to seven. Now, some students quickly write the universal set as one to six. Does that work? No, no, careful! Where is seven? Set C is not a subset of one to six, so that choice fails. Every set must fit inside, right? So, one answer: U equals the set one, two, and so on up to seven. The three dots on screen just mean the numbers in between, okay? Check: set A is inside, set B is inside, and set C is inside. Done. But look, is this the only answer? No! The natural numbers, the capital N with a double line, also contain all three sets. So U equals the natural numbers also works. Any set that holds every set of the discussion is fine. Got it?

5:52Example 2: which of N, W, Z, R can be the universal set

Now, remember our puzzle from the start? Let's finish it, JEE style. Set A is the set of natural numbers x such that x is prime. Set B is the set of integers x such that x is less than zero. Which of these four can serve as a universal set: the natural numbers, the whole numbers, the integers, or the real numbers? On screen they are the capital N, W, Z and R, each with a double line, okay? Pause the video and try it. Done? Okay, let's check. First, write the sets in roster form. Set A: the primes, two, three, five, seven, and so on. I draw it as this circle. Set B: the negative integers, minus one, minus two, minus three, and so on. Every element of set A is a natural number, and every element of set B is negative. So they have no common elements, and their circles are drawn apart, just like we saw. Now, the obvious answer. Set A is made of natural numbers, so the natural numbers must work, right? No, no, careful! This is the trap, the same one from our puzzle. Look at set B again. Minus one, minus two, minus three: are they natural numbers? Are they whole numbers? No! The natural numbers and the whole numbers contain no negative numbers. So set B does not fit inside them, and neither can be the universal set here. Both are out! Now the integers. Every prime here is a natural number, so it is an integer, and every negative integer is an integer, of course. So both circles fit inside the integers, okay? And from the last lesson, you know the integers sit inside the real numbers, so the real numbers hold both sets too. Answer: the integers and the real numbers, both of them. So, our big box from the start? Not the natural numbers, but the integers, or the real numbers. Puzzle solved!

8:02Quick revision: universal set and Venn diagrams

So, quick revision. The universal set contains every set of the discussion as a subset. In a Venn diagram, the universal set is a rectangle, the sets are circles inside it, and the overlaps show common elements. And when you choose the universal set, check that every set fits inside; often more than one choice works. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!