Set Theory

A set is a collection of elements. Here is a set containing all the vowels

Typically the set is denoted by capital alphabet and elements are wrapped up in curly braces and elements are separated using a ‘,’ and the order of the elements does not matter.

Here is a set of even numbers from 0 to 100

We can write the same set in a compact way as well(and this is useful for large sets)

And we read it as ‘E’ is a set of all x’s such that ‘x’ follows the given conditions, in this case, it lies between 0 to 100 and the second condition says that ‘x’ is divisible by 2.

Any element which satisfies the conditions in the compact form of a set is said to belong to the set.

And here is the notation to say that ‘x’ belongs to set ‘S’:

For set ‘E’, 2 belongs to the set and the element 3 does not belong to the set, so we can represent it this way:

Subsets

Say ‘I’ is the set of all integers and now if we defined another set ‘S’ such that it is a set of all negative integers, so it’s obvious that set ‘S’ is completely included inside set ‘I’

Equal sets

If two sets have exactly the equal number of elements, and every element of one set is present in the other set then they are called as equal sets

Universal Set

It is the set that contains all possible things of interest to us, used a lot in the probability theory, and all other sets of interest to us will always be a subset of the universal set. Let’s understand with the help of an example:

Empty Set

An empty set is the complement of the universal set. So, the universal set contains all the elements of the interest, an empty set or a null set contains no elements

Set Operations

Complement Operation

Let’s say the square in the below image represents the universal set which contains all elements of interest and we have this set A(represented by black circle) within the universal set, so all other elements that belong to the universal set but do not belong to ‘A’ form a compliment of ‘A’.

Union operation

So, we have two sets ‘A’(represented by a black circle) and ‘B’(represented by a white circle) in the above image, the union of these two sets is represented by the region in black, in white, and the intersection of the black and the white as highlighted in the below image.

And the union is defined as all elements such that they either belong to A or to B(or condition takes care of the case that they belong to both A and B)

Intersection Operation

The intersection operation gives all elements that belong to set A as well as set B.

We can have these operations on ’n’ sets

So, an element ‘x’ will belong to the union of ‘n’ sets as long as it belongs to anyone of the ‘n’ sets

And an element ‘x’ will said to belong to the intersection of ‘n’ sets only if belongs to all the ‘n’ sets

Properties of set operations

Commutative

If we have two sets A and B, then A union B is the same as B union A and similarly, A intersection B is the same as B intersection A and it follows from the definition as well.

Countable vs Uncountable infinite sets

Usually, we talk of finite sets for example the set of vowels {a, e, i, o, u} is a finite set as it contains finite number of elements, set of all even integers from 0 to 100 also forms a finite set.

We could also have an infinite set and there are two types of infinite sets: countable and uncountable

Let’s say set ‘A’ is the set of all positive even integers and set ‘I’ is the set of all positive integers:

A = {2, 4, 6, 8, ……., }

I = {1, 2, 3, 4, ………,}

We can do the 1–1 mapping between these two sets, we can call the elements as the first element, the second element, third element, and so on and we have enough integers to map every element of the sets. So, the set ‘A’ in this case would be a countable infinite set.

Now a set of real numbers would be something like this:

R = {1, 1.1, 1.11, 1.111, 1.1111, ………}

I = {1, 2, 3, 4, ……………………………}

By the time we reach the element 2 in the ‘R’, we would have exhausted all the elements from ‘I’(assuming we try the 1–1 mapping) as there are infinite real numbers possible between 1 and 2; hence we can not do 1–1 mapping and therefore the set of the real numbers is an uncountable infinite set.

In fact, any set of real numbers even in a small interval say 0 to 1 is going to be an uncountable infinite set whereas the set of all integers up to infinity is going to be a countable infinite set.

References: PadhAI