Distinguish among different types of sets (universal set, equal sets, empty/null set, finite and infinite sets, equivalent sets, and disjoint sets)

Finite and Infinite Sets

A finite set contains a countable number of elements. We can list all elements completely.

Example: $A = \{a, e, i, o, u\}$ — this set has exactly 5 elements.

An empty set, also called a null set or void set, has no elements at all. It is denoted by $\emptyset$ or $\{ \}$.

Example: The set of prime numbers between 31 and 37 is empty because there is no prime number in that range. So $B = \emptyset$.

A singleton set contains only one element.

Example: $C = \{7\}$

An infinite set contains an uncountable number of elements. We represent infinite sets by listing a few elements followed by ellipsis (...).

Example: $D = \{2, 4, 6, 8, ...\}$ — the set of even numbers continues forever.

Universal Set

A universal set is the set that contains all elements of all sets under consideration in a particular problem. It is usually denoted by $\mu$ or $U$.

If we are working with sets of letters and numbers, the universal set would contain both letters and numbers being discussed.

Equal and Equivalent Sets

Two sets are equivalent if they have the same number of elements, regardless of whether the elements are the same. We write $A \equiv B$ or say "A is equivalent to B."

Example: Let $A = \{2, 4, 6, 8\}$ and $B = \{a, b, c, d\}$. Both sets have 4 elements, so $n(A) = n(B) = 4$. Therefore, A is equivalent to B.

Two sets are equal if they contain exactly the same elements, regardless of order. Equal sets must also be equivalent.

Example: If $A = \{1, 2, 3, 4\}$ and $B = \{3, 1, 4, 2\}$, then $A = B$ because all elements of A are in B and all elements of B are in A.

Note: All equal sets are equivalent, but not all equivalent sets are equal.

Disjoint Sets

Two or more sets are disjoint if they have no elements in common. Their intersection is the empty set.

Example: Let $A = \{1, 3, 5, 7\}$ (odd numbers less than 10) and $B = \{2, 4, 6, 8\}$ (even numbers less than 10). Sets A and B share no common elements, so they are disjoint. We write $A \cap B = \emptyset$.

If sets have common elements, they are called joint sets.