1. Ordered Pair
An ordered pair is a pair of elements written as $(a,b)$ where the order matters: $(a,b) \neq (b,a)$ unless $a=b$.
Example: $(1,2)$ is different from $(2,1)$.
2. Cartesian Product
The Cartesian product of two sets $A$ and $B$ is $A \times B = \{(a,b) : a \in A, b \in B\}$.
Example: If $A = \{1,2\}, B = \{x,y\}$, then $A \times B = \{(1,x),(1,y),(2,x),(2,y)\}$.
3. Relation
A relation from set $A$ to set $B$ is a subset of $A \times B$.
Example: If $A = \{1,2\}, B = \{x,y\}$, a relation $R = \{(1,x),(2,y)\} \subseteq A \times B$.
4. Domain and Range
Domain: The set of all first elements of ordered pairs in a relation $R$, denoted $dom(R)$.
Range: The set of all second elements, denoted $ran(R)$.
Example: For $R = \{(1,x),(2,y)\}$, $dom(R) = \{1,2\}, ran(R) = \{x,y\}$.
5. Types of Relations
- Void Relation: $R = \varnothing$, no elements.
- Universal Relation: $R = A \times B$, contains all possible pairs.
- Identity Relation: $I = \{(a,a) : a \in A\}$
- Reflexive Relation: For all $a \in A$, $(a,a) \in R$
- Symmetric Relation: If $(a,b) \in R \implies (b,a) \in R$
- Transitive Relation: If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$
- Equivalence Relation: Relation that is reflexive, symmetric, and transitive
6. Examples of Relations
Let $A = \{1,2,3\}$, $B = \{x,y\}$:
- Void Relation: $R = \varnothing$
- Universal Relation: $R = \{(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)\}$
- Identity Relation (on $A$): $I = \{(1,1),(2,2),(3,3)\}$
- Reflexive Relation (on $A$): $R = \{(1,1),(2,2),(3,3),(1,2)\}$
- Symmetric Relation: $R = \{(1,2),(2,1),(2,3),(3,2)\}$
- Transitive Relation: $R = \{(1,2),(2,3),(1,3)\}$
- Equivalence Relation: $R = \{(1,1),(2,2),(3,3),(1,1)\}$