Plus Two Maths Notes Chapter 1 Relations and Functions is part of Plus Two Maths Notes. Here we have given Plus Two Maths Notes Chapter 1 Relations and Functions.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Notes |

Chapter | Chapter 1 |

Chapter Name | Relations and Functions |

Category | Kerala Plus Two |

## Kerala Plus Two Maths Notes Chapter 1 Relations and Functions

**Different Types of Relations**

**Empty relation:**

The relation R in A given by R = ϕ ⊂ A × A**Universal relation:**

The relation R in A given by R = A × A**Reflexive relation:**

A relation R in A with (a, a) ∈ A, ∀a∈A**Symmetric relation:**

A relation R in A satisfying (a,b)∈ R ⇒ (b, a)∈ R, ∀a,b∈A**Transitive relation:**

A relation R in A satisfying (a, b)∈ R and (b, c)∈ R ⇒ (a,c)∈ R, ∀a,b,c∈A**Equivalence relation:**

A relation R in A which is reflexive, symmetric and transitive.**Equivalence class [a] containing a∈A:**

For an equivalence relation R in A is the subset of A containing all elements b related to a.

**Different Types of Functions**

**One-one (Injective) function:**

A function ƒ: X → Y is one-one if ƒ(x_{1})= A(x_{2}) z> x_{1 }= x_{2}, ∀x_{1}, x_{2}∈ X**Many-one function:**

A function which is not one-one.**Onto (Surjective) function:**

A function ƒ: X → Y is onto if given any y∈Y, ∃x∈X such that ƒ(x) = y.- ƒ: X → Y is onto if and only if range of ƒ = Y
**Bijective function:**

A function ƒ: X → Y is bijective 1ff is both one-one and onto.**Composition of functions:**

ƒ: A → B and g: B → C is the function gof: A → C given by (gof)(x)=g(ƒ(x)).- A function ƒ: X → Y is invertible if there exists a function g: Y → X such that gof = I
_{x}and fog = I_{y} - A function ƒ: X → Y is invertible if and only if ƒ is one-one and onto (bijective).

**Binary Operations**

- A binary operation * on a set A is a function*: A × A → A
- An element e ∈ A in the identity element for if a*e e*a = a, ∀a∈A
- An element a ∈ A is invertible for * there exists b ∈ A such that a*b = b * a e, then b = a
^{-1}. - * is commutative if a * b = b * a, ∀ a, ∈b ∈ A
- * is associative if (a * b) * c = a* (b * c) ∀ a, b, c ∈ A.

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