Subsets, Intervals and Power Set

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. I take a tiny set with just three letters in it: a, b and c, that's all. How many subsets does it have? Three? Six? Oh, I hear these answers in every class, and both are wrong! By the end of this video, you will count them in two seconds. And look, JEE loves this topic, because it is full of small traps. So, let's begin.

0:30Subset: every element of A is in B

So, look at the picture. Set A sits completely inside set B, right? Every element of A is also in B, and we say A is a subset of B, okay? The sign looks like a rounded less than or equal to. In symbols: if x is in A, then x is in B, okay? This curvy E means is an element of, the single arrow means implies, and the double arrow means if and only if. Clear? Now, if A is a subset of B, but not equal to B, then we call it a proper subset. We write it with a small slash under the sign: subset, but not equal, okay? Many books simply write it without the line below, so don't get confused. And from the other side, B is a superset of A, with the same sign turned around.

1:19Basic facts about subsets

Now, two facts that JEE loves. Every set is a subset of itself, obviously. But tell me: is the empty set, this circle with a slash, a subset of every set? Yes, always! To fail, it would need an element that is not in A, but it has no elements at all, you see? So it can never fail. Completely logical. Next, a question. How do you prove that two sets are equal? Show it both ways! First, A is a subset of B, and then, B is a subset of A, okay? If each one sits inside the other, they must be the same set. This is the standard method, so remember it. And subsets form chains. If A is inside B, and B is inside C, then A is inside C, okay? Like our number systems: natural numbers, whole numbers, integers, rationals, reals and complex numbers. Each one sits inside the next.

2:22The trap: element of versus subset of

Now, the most common trap in this chapter. The curvy E means element of: it connects an element with a set. The subset sign connects a set with a set. Got it? Take this set A, with only two elements: the number one, and the set two, three, as one single packet. Is the packet two, three an element of A? Yes, of course! It is sitting right there in the list. So this one is true. Now, is it also a subset of A? Obviously yes, right? No, no, careful! This is the trap. For a subset, every element of two, three must lie in A, okay? Is two in A? No! The elements are only one, and the packet. The crossed sign means not an element of. So this is false. But add one more pair of curly brackets. Now the only element is the packet two, three, and that is in A, so this one is true. So, always ask: element with a set, or set with a set?

3:25Intervals: subsets of the real line

Alright, now intervals: subsets of the real numbers, on the number line. Round brackets, a comma b, means all real x between a and b, with the ends excluded: the hollow circles. Square brackets include the ends: the solid dots. Open and closed, with me? Mix them, one square and one round, and you get the semi-open intervals. And the length? Same for all four: b minus a, okay? Intervals can also run to infinity, this sideways eight. Square bracket a, comma infinity, is the set of all x such that x is greater than or equal to a, okay? And the whole real line, R, is minus infinity to infinity. Now tell me: can I put a square bracket at infinity? Never! Infinity is not a number, so it can never be included. Always a round bracket there, okay?

4:21Counting subsets: take it or leave it

Now, back to our puzzle: how do we count subsets without listing them? Here is the topper's way. If A has n elements, go element by element and decide: take it, or leave it. Two choices. Two choices, n times. So how many subsets? Yes! Two into two into two, n times: two to the power n, that's it! Proper subsets? Every subset except A itself. So two to the n, minus one. Okay? And non-empty proper subsets? Why minus two now? Exactly! We throw out A itself, and the empty set. So two to the n, minus two.

5:03Power set: the set of all subsets

Now, put all the subsets of A into one big set: that is the power set, written with this curly P, okay? Same counting, so if A has n elements, the power set has two to the n elements. Now a tricky one. How many elements in the power set of the empty set? Zero, right? No! Careful. Look, the empty set has exactly one subset: the empty set itself. So its power set is the empty set inside curly brackets, and it has one element. It is not empty! Two to the power zero is one, so the formula agrees. It fits perfectly.

5:43Example 1: all 8 subsets of {a, b, c}

Alright, now let's finish our puzzle. Write all the subsets of the set a, b, c, and count the proper subsets. Pause the video, and try it first. Done? Okay, let's check. Go by size. No element: the empty set. Don't forget it! One element: a, b, c, three of them. Two elements: a b, b c, a c, three more. And all three together. Now count: one, plus three, plus three, plus one. Eight! And two to the power three is also eight. So three and six were wrong, right? The answer to our puzzle is eight. Proper subsets: all except the full set, so eight minus one, seven. Exactly two to the n, minus one. Really simple.

6:38Example 2: subsets of A that meet B, count the opposite

Now a harder one, a JEE Main question from twenty twenty two. Set A has the numbers one to seven, and set B has three, six, seven and nine. How many subsets C of A meet B? That means C intersection B is not equal to the empty set. Pause and try. Hint: count the opposite. Done? Okay, let's check. Counting directly means too many cases. So, here is the topper's trick: count the opposite. Count the subsets that miss B completely, then subtract them from the total. Now, B has four elements, so we remove all four from A, and count the subsets of the three that are left, right? No, no, careful! That gives one hundred and twenty, and that is wrong. Is nine an element of A at all? No! Inside A, only three, six and seven belong to B: that is A intersection B, the part they share. Only this part matters, okay? Any subset that avoids three, six and seven can use only one, two, four and five. Four elements, so two to the four: sixteen. And A has seven elements, so two to the seven: one hundred and twenty eight subsets in all. Minus sixteen: one hundred and twelve! See? No case work at all. Just count the opposite.

8:09Quick revision: element vs subset, 2^n subsets, intervals

So, quick revision. Element of connects an element with a set; subset of connects a set with a set. If a set has n elements, it has two to the n subsets, two to the n minus one proper subsets, and its power set has two to the n elements. In intervals, a square bracket includes the end, a round bracket excludes it, and infinity always gets a round bracket. Now go and try the practice problems on this page. You will enjoy them, I promise. All the best!