Class 12 Relation and Function Lec-06

The Relations and Functions unit builds the foundation for higher algebra and calculus by describing how mathematical objects pair and interact with one another. Core Concepts Cartesian Product ($A \times B$)The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$ .Example: If $A = \{1, 2\}$ and $B = \{x, y\}$, then $A \times B = \{(1,x), (1,y), (2,x), (2,y)\}$.RelationsA relation is simply any subset of $A \times B$. It establishes a connection between elements of two sets. Domain: Set of all first elements in the ordered pairs. Range: Set of all second elements actually matched in the relation. Types of Relations: Reflexive: Every element relates to itself ($aRa$).Symmetric: If $aRb$, then $bRa$.Transitive: If $aRb$ and $bRc$, then $aRc$.Equivalence Relation: A relation that is reflexive, symmetric, and transitive simultaneously. Functions A function is a special type of relation where every input has exactly one output. No element in the domain can map to multiple values in the target set (codomain).Key Types of Functions One-to-One (Injective): Distinct inputs produce distinct outputs .Onto (Surjective): Every element in the codomain has at least one input mapping to it (Range = Codomain).Bijective: Both One-to-One and Onto (necessary condition for a function to have an inverse).