CBSE Class 12 Mathematics Relation And Function Worksheet Set 03

TYPES OF RELATIONS:

• A relation \( R \) in a set \( A \) is called reflexive if \( (a, a) \in R \) for every \( a \in A \).
• A relation \( R \) in a set \( A \) is called symmetric if \( (a_1, a_2) \in R \) implies that \( (a_2, a_1) \in R \), for all \( a_1, a_2 \in A \).
• A relation \( R \) in a set \( A \) is called transitive if \( (a_1, a_2) \in R \) and \( (a_2, a_3) \in R \) together imply that \( (a_1, a_3) \in R \), for all \( a_1, a_2, a_3 \in A \).

EQUIVALENCE RELATION

• A relation \( R \) in a set \( A \) is said to be an equivalence relation if \( R \) is reflexive, symmetric and transitive.

Equivalence Classes

• Every arbitrary equivalence relation \( R \) in a set \( X \) divides \( X \) into mutually disjoint subsets \( (A_i) \) called partitions or subdivisions of \( X \) satisfying the following conditions:
• All elements of \( A_i \) are related to each other for all \( i \).
• No element of \( A_i \) is related to any element of \( A_j \) whenever \( i \neq j \).
• \( A_i \cup A_j = X \) and \( A_i \cap A_j = \Phi \), \( i \neq j \). These subsets \( (A_i) \) are called equivalence classes.
• For an equivalence relation in a set \( X \), the equivalence class containing \( a \in X \), denoted by \( [a] \), is the subset of \( X \) containing all elements \( b \) related to \( a \).

Function: A relation \( f: A \rightarrow B \) is said to be a function if every element of \( A \) is correlated to a unique element in \( B \).

* \( A \) is domain

* \( B \) is codomain

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