Example 21 - Chapter 1 Class 12 Relation and Functions
Last updated at April 16, 2024 by Teachoo
Function - Definition
Chapter 1 Class 12 Relation and Functions
Concept wise
- Relations - Definition
- Empty and Universal Relation
- To prove relation reflexive, transitive, symmetric and equivalent
- Finding number of relations
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Function - Definition
- To prove one-one & onto (injective, surjective, bijective)
- Composite functions
- Composite functions and one-one onto
- Finding Inverse
- Inverse of function: Proof questions
- Binary Operations - Definition
- Whether binary commutative/associative or not
- Binary operations: Identity element
- Binary operations: Inverse
Transcript
Example 21 Let f : X → Y be a function. Define a relation R in X given by R = {(a, b): f(a) = f(b)}. Examine whether R is an equivalence relation or not. Equivalence relation are Relations which are reflexive, transitive and symmetric. R = {(a, b): f(a) = f(b)} Check reflexive Since f (a) = f (a), ∴ (a, a) ∈ R, Hence, R is reflexive. Check symmetric If f (a) = f (b), then f (b) = f (a) Hence, (b, a) ∈ R So, if (a, b) ∈ R , then (b, a) ∈ R. ∴ R is symmetric. Check transitive If (a, b) ∈ R ∴ f(a) = f(b) Also if, (b, c) ∈ R ∴ f(b) = f(c) From (1) & (2) f(a) = f(c) ∴ (a, c) ∈ R, ∴ If (a, b) ∈ R & (b, c) ∈ R , then (a, c) ∈ R ∴ R is transitive. Hence, R is an equivalence relation.
