A set is a collection of distinct objects. Let’s look at some examples:

- A family. Mother, father and two children forms a simple set of objects: a set of four people. Each person differs from other persons.
- A deck of cards. A standard deck of cards has 52 cards inside and each card differs from each other. It forms a set of 52 distinct objects.
- A set of all natural numbers forms our first infinite set. Each number differs from other numbers.

We usually denote the set as follows: *{1, 2, 3, 4}*. This forms a set of four numbers. We can write down the three sets defined above as follows:

*{mother, father, son, daughter}**{two of clubs, three of clubs, …, red ace}**{1, 2, 3, 4, 5, …}*

We use the three dots to denote an interval that is quite clear.

## Elements of a Set

Each set consists of *elements* or *members*. Let’s suppose we have an object *x* and a set *S*. The object *x* is either an element of the set *S* or it isn’t element of the set *S*. If the object *x* is member of the set *S* we denote it as *x ∈ S*. If the object *x* doesn’t belong to the set *S* we denote it as *x ∉ S*. This “being element of” membership is all we need for defining a set.

Let’s says that for our set *S* we have *1 ∈ S*, *2 ∈ S* and *3 ∈ S* and that’s all. We can write down the set using the curly brackets syntax: *S = {1, 2, 3}*. From our definition it follows:

- The order doesn’t matter.
*{1, 2, 3}*is the same set as the set*{2, 3, 1}*. - The repetition doesn’t matter.
*{1, 2, 3}*is the same set as*{1, 2, 2, 3, 3, 3}*. Even though the number 3 is there three times it still tells us that*3 ∈ S*and nothing more. In the definition of “being element of a set” are no words about the count. - The set can be empty. We can have a set
*S = {}*. Sometimes we use different symbol for the empty set:*∅*.

## What is Subset of a Set

We can have two sets: *A* and *B*. We say that the set *A* is a subset of a set *B* if all elements from the set *A* are also included in the set *B*. We denote this relation as *A ⊆ B*. Examples:

- The set
*A = {1, 2}*is a subset of*B = {1, 2, 3}*since both the elements*1*and*2*from*A*are also in*B*. On the other hand, the set*B*is not a subset of*A*since the element*3*is not member of the set*A*. Thus we have*A ⊆ B*but we have*B ⊈ A*. - The set
*A = {a, b, c}*is not a subset of*B = {a, b, e}*since the element*c*is not member of set*B*. - The empty set
*∅*is a subset of each set. So for each set*A*we have*∅ ⊆ A*. - The set
*A*is always a subset of a set*A*. The set is its own subset. It is simply because the set*A*contains all the elements that contains the set*A*. Obviously.

We can express a subset of a set in a picture. Let’s suppose two sets *A* and *B* with relation *A ⊆ B*. Then we can express this relation as follows:

We have a “bigger” set *B* and inside of it is a “smaller” set *A*. This represents the fact that the set *A* contains all elements from the set *B* and the set *B* can contains some additional elements.

Remark: Sometimes we use another symbol for the subset relation. Instead of *⊆* we use *⊂*. And sometimes the latter symbol is used for *proper subset*.

The set *A* is a **proper subset** of *B* if *A ⊆ B* but *B ⊈ A*. Basically it means that the sets must not be equal, the set *B* must contain some elements that the set *A* doesn’t contain. Examples:

- The set
*A = {1, 2}*is a proper subset of*B = {1, 2, 3}*, we can write*A ⊂ B*. - The set
*A = {1, 2, 3}*is not a proper subset of*B = {1, 2, 3}*since the set*A*contains all the elements of*B*, thus*A ⊄ B*.

## Set Equality

With the knowledge of the term *subset* we can easily define the set equality. Two sets *A* and *B* are equal if and only if *A ⊆ B* and *B ⊆ A*. Does it make sense?

- If
*A ⊆ B*then all elements from*A*are also in*B*, - if
*B ⊆ A*then all elements from*B*are also in*A*.

Thus, the only conclusion is that both sets contains the exact same elements. Then we say the sets are equal and we write *A = B*. Examples:

*{1, 2, 3}*=*{1, 2, 3}*(trivially)*{1, 2}*=*{1, 1, 1, 1, 2}*(because sets doesn’t allow element repetition)*{a, b, c} ≠ {a, b}*(because the second set doesn’t contain the*c*element)*{{1}, 2} ? {1, 2}*– this one is tricky. Do the sets equal? The answer is no, because the first set contains as an element the set*{1}*and the seconds set contains as an element the number*1*. Those are not equal, the set*{1}*differs from the number*1*simply because any set cannot be equal to any number. A set is a set and a number is a number. Thus*{{1}, 2} ≠ {1, 2}*.

## Power Set

The power set of a set *S* is a set of all subsets of *S*. Huh, that sounds complicated! Let’s look at the example. Suppose we have set *S = {a, b}*. What are all the subsets of this set *S*?

*∅*since empty set is subset of each set,*{a}*and*{b}*,*{a, b}*since*S*is always subset of*S*.

Now we put all those subsets to a new set. We get:

And this is exactly the power set of *S*. We usually denote it as *P(S)*. Thus

Some basic properties for the power set of any set *S*:

*∅ ∈ P(S)*: the empty set is always included in the power set.*S ∈ P(S)*: the set*S*itself is always included in the power set.