The Notion of Sets

1. Empty Set: A set with no elements, denoted by $\{ \}$ or $Ø$. Example: The set of Natural numbers between 1 and 2.

2. Finite Set: A set with a specific, countable number of elements. Example: The total number of peoples in Ethiopia, The set of colors in a rainbow.

3. Infinite Set: A set with an unlimited number of elements. Example: The set of natural numbers $\{1, 2, 3, \dots\}$.

4. Power Set: The set of all possible subsets of a given set, including the empty set and the set itself. Example: The power set of $\{a, b\}$ is $\{ \{\}, \{a\}, \{b\}, \{a, b\} \}$.

5. Subset: Set A is a subset of set B if every element of A is also in B, denoted as $A \subseteq B$. Example: $\{red, green\}$ is a subset of $\{red, green, blue\}$.

6. Proper Subset: A subset that contains some but not all elements of another set. Set A is a proper subset of set B if $A \subset B$ and $A ≠ B$. Example: $\{red, green\}$ is a proper subset of $\{red, green, blue\}$.

Set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵. Mathematically, we write this as 𝐴 ⊆ 𝐵.

• Any set is a subset to itself.
• Empty set is a sub set of every set.
• If set 𝐴 is finite with 𝑛 elements, then the number of subsets of set 𝐴 is $2^𝑛$

If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called the proper subset of 𝐵 and it can be written as 𝐴 ⊂ 𝐵.

• For any set 𝐴, 𝐴 is not a proper subset to itself.
• If set 𝐴 is finite with 𝑛 elements the number of proper subsets of set 𝐴 is $2^{𝑛 − 1}$.
• Empty set is a proper subset of any other sets.