1. Subsets

1.1 Definition of a subset

A set \(A\) is a subset of a set \(B\) if every element of \(A\) is also an element of \(B\).

The mathematical notation is:

This is read as “A is a subset of B.”

Example 1

Consider the sets:

Every element of \(A\) appears in \(B\).

Therefore:

However, the reverse is not true:

This is because the elements \(3\) and \(4\) belong to \(B\) but do not belong to \(A\).

1.2 Important subset rules

Rule 1: Every set is a subset of itself.

Rule 2: The empty set is a subset of every set.

Rule 3: A subset may contain all the elements of another set.

For example:

Since the order of elements does not matter in a set:

Therefore:

1.3 Subset versus element

Students often confuse the symbols \(\in\) and \(\subseteq\).

Consider:

Then:

This means that \(2\) is an element of \(A\).

Also:

This means that the set containing \(2\) is a subset of \(A\).

But:

because the elements of \(A\) are the numbers \(1,2,3\), not the set \(\{2\}\).

Table – Subset versus element

Notation	Meaning	Example
\(x\in A\)	x is an element of A	\(2\in\{1,2,3\}\)
\(x\notin A\)	x is not an element of A	\(5\notin\{1,2,3\}\)
\(A\subseteq B\)	A is a subset of B	\(\{1,2\}\subseteq\{1,2,3\}\)
\(A\nsubseteq B\)	A is not a subset of B	\(\{1,5\}\nsubseteq\{1,2,3\}\)

2. Proper Subsets

A set \(A\) is a proper subset of \(B\) if:

• Every element of \(A\) belongs to \(B\).

• At least one element of \(B\) does not belong to \(A\).

The notation is:

Equivalently:

Example 2

Because all elements of \(A\) are in \(B\), but \(A\ne B\):

Example 3

Here:

Therefore:

but:

Important: A proper subset cannot be equal to the original set.

2.1 Number of subsets

If a finite set has \(n\) distinct elements, then the number of subsets is:

The number of proper subsets is:

Example 4

Find the number of subsets and proper subsets of:

Solution

There are three elements:

Number of subsets:

All subsets are:

Number of proper subsets:

The only subset that is not proper is \(A\) itself.

Remember: The empty set is included when counting subsets.

3. Equal Sets

Two sets are equal if they contain exactly the same elements.

The notation is:

Example 5

Although the order is different, the sets contain the same elements.

Therefore:

Repeated elements do not change a set:

3.1 How to prove that two sets are equal

Two sets are equal if and only if each is a subset of the other:

This is called the double-inclusion method.

4. Universal Set and Empty Set

4.1 Universal set

The universal set contains all elements under consideration in a particular problem.

It is commonly represented by:

Example 6

Suppose we are studying whole numbers from \(1\) to \(10\).

Define:

Here, \(A\) contains the even numbers in \(U\).

Thus:

The universal set depends on the problem. It does not always mean all real numbers.

4.2 Empty set

The empty set is a set containing no elements.

It is represented by:

Its cardinality is:

For example, consider the set of real numbers satisfying:

No real number satisfies this equation.

Therefore, the solution set is:

Important distinction

The first set contains zero elements. The second contains one element: the empty set.

5. Disjoint Sets

Two sets are disjoint if they have no elements in common.

Mathematically:

Example 7

There are no common elements.

Therefore:

Disjoint sets are useful when classifying objects into categories that do not overlap.

Part II – Set Operations

6. Union of Sets

The union of two sets contains all elements that belong to \(A\), to \(B\), or to both.

The notation is:

In set-builder notation:

The word “or” is inclusive: an element may belong to either set or both.

Example 8

Combining all distinct elements gives:

The number \(3\) appears only once because sets do not contain duplicate elements.