Types of Sets

Transcript

0:00Introduction

Hello everyone! Let's start with one small puzzle. Set one: all even prime numbers greater than two. Set two: all real roots of x squared plus one equals zero. One is about primes, the other about an equation. Totally different, right? So tell me, can these two sets be equal? Keep your answer in mind. By the end of this video you will know for sure, and you will see why JEE loves to set traps on this topic. So, let's begin.

0:34Empty set and singleton set

First, the simplest set of all: the empty set. It has no element at all. It is also called the null set, or the void set. We write it as a circle with a slash through it, or as two curly brackets with nothing between them. Same thing. Clear? Now look at this set. The double struck R means real numbers, the small curved E means belongs to, and the colon is read as such that. So tell me, which real number, when squared, gives minus one? None! Because the square of a real number is never negative. So this set is the empty set. Next, the singleton set. Singleton means single: the set has exactly one element. Not zero, not two, just one. Really simple. For example, the set of all even primes. How many even primes do you know? Just one, the number two! Every other even number can be divided by two, so it is not prime. So this set is a singleton.

1:36Finite and infinite sets

Alright, now finite sets. We call a set finite if it is empty, or if its elements can be counted and the counting ends. Quick question: is the empty set finite or infinite? Finite! Read the line again: empty counts as finite. Many students miss this, okay? For a finite set, the number of elements is called its cardinal number. We write it as n of A, or as A between two vertical bars. Same meaning, got it? And a set that is not finite is infinite: the counting never ends. Like the natural numbers, this double struck N: one, two, three, forever. The points on a line. And the open interval from zero to one, in round brackets. See, it looks small, but it holds infinitely many real numbers, right?

2:29Equal and equivalent sets

Now let's compare two sets. Two sets A and B are equal, written A equals B, if they have exactly the same elements. Nothing extra on either side. Equivalent is a different word, so careful. Two sets are equivalent if n of A equals n of B: the same number of elements. The elements themselves may differ. Now, a question that JEE loves. If two sets have the same number of elements, are they equal? Same count, same set, right? No, no, careful! That is the trap. Look, let's get the direction right. If two sets are equal, they have the same elements, so of course the same count. So equal sets are always equivalent. No exceptions. But not the other way round. Take the set one, two, three, and the set of letters a, b and c. Three elements each, so they are equivalent. But the number one is not the letter a, right? So they are not equal. Got it?

3:32There is only one empty set

Now, back to our puzzle. Two is the only even prime, so there is no even prime greater than two. Empty! And x squared plus one equals zero needs x squared equal to minus one. No real number does that. So the second set is empty too. And here is the key fact: there is only one empty set. Both sets have exactly the same elements: nothing. So they are equal! One about primes, one about an equation, and still the same set. Beautiful, isn't it?

4:02Disjoint sets: nothing in common

One more type, about two sets together. Look at these two circles, A and B, okay? The highlighted middle part holds their common elements. Two sets are called disjoint when they have no common element at all. So for disjoint sets, there is no middle part; the circles sit apart. This cap shaped symbol means intersection, the common part. So, A intersection B equals the empty set. Clear?

4:30JEE trap: three sets that look alike

Now the most famous trap of this topic. The empty set has no element, so its n is zero. Fine. Now look at zero inside curly brackets, and the empty set symbol inside curly brackets. They look like nothing too, don't they? So these are also empty, right? No, no! Let's check. The set containing zero is not empty. Zero is a number, sitting inside the brackets as an element. So this set has one element. Its n is one: a singleton! And the set containing the empty set? Think of a box with an empty box inside. The outer box is not empty; it holds one thing, the empty box. So its n is one again. So, three different sets, with counts zero, one and one. Remember, never call those last two empty in the exam. Okay?

5:23Example 1: empty, singleton, finite or infinite

Now, example one. Classify each set as empty, singleton, finite or infinite. One: natural numbers strictly between one and two. Two: lines through a given point. Three: natural numbers whose square is nine. Four: real numbers between zero and one. Five: natural numbers that are multiples of five, less than one hundred. Pause the video, and try it first. Done? Okay, let's check. Part one: is there any natural number between one and two? No! So it is the empty set. Parts two and four together. Through a given point you can draw endless lines. And between zero and one there are endless real numbers. So both sets are infinite. Now part three. x squared equals nine, so x is three or minus three. Two elements, right? Wait, careful! Read the question again. The question says x is a natural number. Minus three is not natural. On screen, the belongs to sign with a slash through it means, is not an element of: minus three is not an element of N, okay? So only three survives, and the set is a singleton. Part five: five, ten, fifteen, and so on, that is what the dots mean, up to ninety five. Ninety five divided by five is nineteen. Nineteen elements, and the counting ends, so it is finite. Perfectly clear.

7:00Example 2: find n(A) and test equivalent sets

Now a harder one, JEE level. The double struck Z means the integers. Set A: integers x with x squared minus x minus six less than or equal to zero. Set B: natural numbers that divide thirty two. Find n of A, and decide: are A and B equivalent? Pause and try. Hint: factorise first. Done? Let's check. x squared minus x minus six is x minus three, times x plus two. When is this product zero or negative? Between the two roots, and including them too. This arrow means implies. So x lies from minus two to three, both ends included, okay? But x must be an integer: minus two, minus one, zero, one, two, three. Don't forget zero and the negatives! Six elements, so n of A is six. Now set B: the divisors of thirty two are one, two, four, eight, sixteen and thirty two. Six again, so n of B is six. Six and six. So the two sets are equal, right? Hmm, remember the trap we just learnt? The same count only makes them equivalent. Minus two is in A but not in B, so they are not equal. Equivalent, but not equal. See how the trap tries to catch you?

8:32Quick revision: empty, singleton, equal vs equivalent

So, quick revision. Empty: no element. Singleton: exactly one element. Equal sets have the same elements; equivalent sets just have the same number of elements. And the empty set, the set containing zero, and the set containing the empty set are three different sets. Now try the practice problems on this page. You will find them easy, I promise. All the best!