Operations Examples

Basic Set Operations

1. Union (A ∪ B)

The union of two sets $A$ and $B$ is the set of all elements that are in $A$ OR $B$ OR both.

Example:
A = {1, 2, 3}, B = {3, 4, 5}

$A \cup B = \{1, 2, 3, 4, 5\}$

2. Intersection (A ∩ B)

The intersection is the set of elements common to both $A$ AND $B$.

Example:
A = {1, 2, 3}$,$B = {3, 4, 5}

$A \cap B = \{3\}$

3. Difference (A - B)

The difference is the set of elements in $A$ but not in $B$.

Example:
A = {1, 2, 3}, B = {3, 4, 5}

$A - B = \{1, 2\}$

$B - A = \{4, 5\}$

4. Symmetric Difference (A △ B)

The symmetric difference is the set of elements in either $A$ or $B$, but not in both.

Example:
A = {1, 2, 3}, B = {3, 4, 5}

$A \triangle B = \{1, 2, 4, 5\}$

5. Complement (A′ or Aᶜ)

The complement of $A$ is the set of all elements not in $A$, relative to a universal set $U$.

Example:
Let U = {1, 2, 3, 4, 5}, $A = \{2, 4\}$

$A' = U - A = \{1, 3, 5\}$

6. Subsets

A set $A$ is a subset of $B$ ($A \subseteq B$) if every element of $A$ is also in $B$.

Example:
$A = \{1, 2\}$, $B = \{1, 2, 3\}$

$A \subseteq B$

7. Powerset (P(A))

The powerset of $A$ is the set of all subsets of $A$.

Example:
$A = \{a, b\}$

$\mathcal{P}(A) = \{\varnothing, \{a\}, \{b\}, \{a, b\}\}$

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Each operation helps us analyze relationships between sets, such as finding common elements (intersection), combining sets (union), or exploring all possible groupings (powerset).