All other sets in that context are subsets of U.

Graphical representation of relationships between sets using rectangles (for U) and circles (for subsets).

Union of Sets ($A \cup B$): The set of all elements which are either in A or in B (including those in both).
- $A \cup B = \{x : x \in A \text{ or } x \in B\}$
- Properties: Commutative ($A \cup B = B \cup A$), Associative ($(A \cup B) \cup C = A \cup (B \cup C)$), Identity ($A \cup \phi = A$), Idempotent ($A \cup A = A$), Law of U ($U \cup A = U$).
Intersection of Sets ($A \cap B$): The set of all elements which are common to both A and B.
- $A \cap B = \{x : x \in A \text{ and } x \in B\}$
- Disjoint Sets: If $A \cap B = \phi$.
- Properties: Commutative ($A \cap B = B \cap A$), Associative ($(A \cap B) \cap C = A \cap (B \cap C)$), Law of $\phi$ and U ($\phi \cap A = \phi, U \cap A = A$), Idempotent ($A \cap A = A$), Distributive ($A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$).
Difference of Sets ($A - B$): The set of elements which belong to A but not to B.
- $A - B = \{x : x \in A \text{ and } x \notin B\}$
- Note: $A - B \neq B - A$.
- Sets $A-B$, $A \cap B$, and $B-A$ are mutually disjoint.

Definition: Let U be the universal set and A a subset of U. The complement of A is the set of all elements of U which are not elements of A.
$A' = \{x : x \in U \text{ and } x \notin A\} = U - A$.
Properties:
- Complement laws: $A \cup A' = U$, $A \cap A' = \phi$.
- De Morgan's laws: $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$.
- Law of double complementation: $(A')' = A$.
- Laws of empty set and universal set: $\phi' = U$, $U' = \phi$.

For finite sets A and B:
- If $A \cap B = \phi$, then $n(A \cup B) = n(A) + n(B)$.
- $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
- $n(A - B) = n(A) - n(A \cap B)$.
For finite sets A, B, and C:
- $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$.

Definition: For non-empty sets P and Q, the Cartesian product $P \times Q$ is the set of all ordered pairs $(p, q)$ where $p \in P$ and $q \in Q$.
- $P \times Q = \{(p, q) : p \in P, q \in Q\}$.
If $P$ or $Q$ is empty, then $P \times Q = \phi$.
Two ordered pairs $(a, b)$ and $(x, y)$ are equal iff $a=x$ and $b=y$.
If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$.
If A or B is infinite, then $A \times B$ is infinite.
$A \times A \times A = \{(a, b, c) : a, b, c \in A\}$ is an ordered triplet.
Generally, $A \times B \neq B \times A$.
$R \times R = \{(x, y) : x, y \in R\}$ represents points in 2D plane.
$R \times R \times R = \{(x, y, z) : x, y, z \in R\}$ represents points in 3D space.