Plus One Maths Notes Chapter 2 Relations and Functions is part of Plus One Maths Notes. Here we have given Kerala Plus One Maths Notes Chapter 2 Relations and Functions.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus One |

Subject | Maths Notes |

Chapter | Chapter 2 |

Chapter Name | Relations and Functions |

Category | Plus One Kerala |

## Kerala Plus One Maths Notes Chapter 2 Relations and Functions

**Cartesian Product of Sets**

**Ordered pair**. A pair of elements grouped together in a particular order,

i.e., (a,b), a ∈ A and b ∈ B- In general (a, b) ≠ (b, a)
- If (a, b) = (x, y), then a = x and b = y
- Cartesian Product. A × B = {(a, b) : a ∈ A and b ∈ B}
- In general, A× B ≠ B ×A

A × B = B × A, if A = B - If n(A) = p and n(B) = q, then n(A x B) = pq
- n(A × B) = n(B × A)
- If A ≠ ϕ or B≠ ϕ , then A x B = ϕ
- A ≠ ϕ, B ≠ ϕ and either A or B is an infinite set, then A × B is an infinite set
- Given set A, A
^{2}= A ×A Graphical representation, ordered pair (x, y), x ∈ ×, y ∈ Y .

**Relation**

- R is a relation from A to B if R⊆A x B

- Universal relation from A to B.

A X B itseIf - Empty relation, the null set ϕ
- A relation in A is a subset of A X A
- If n(A) = p, n(B) = q, then the total number of relations from A to B = 2
^{pq} - If R: A → B,
- Co-domain = B
- Domain = {x ∈ A / (x, y) ∈ R for some y ∈ B}
- Range = {y ∈ B / (x, y) ∈ R for some x ∈A}

- Range of R ⊆ Co-domain of R
- If (x, y) ∈R, then image of x = y
- Inverse relation : If R = {(a, b) / a ∈ A, b ∈ B, aRb}then R
^{-1}= {(b, a): (a, b) ∈R}

**Functions**

- A function T from A to B : A relation in which each element of set A is related to unique element of set B.
- If f: A → B and (x, y) ∈ f, then f(x) = y
- The image of x = y
- The pre-image of y = x
- Domain of f={x ∈ A /f(x) ∈ B}
- Co-domain of f = B
- The range of f, f(A)={f(x)/x ∈ D(f)}

- The number of functions from A to B = (n(B))m
^{n(A)} - If range of f ⊆ R, then f is a real valued function.
- If D(f) ⊆ R, range ⊆ R, then f is a real function
**Graph of a real function:**The set of all order ed pairs (x, f(x)) in the X-Y plane.- Identity function is

f: R→ R, f(x) = x, ∀ x ∈R **Constant function is f:**R → R, f (x) = k, ∀ x ∈R, where k is a constant Modulus

**Greatest integer function (step function) is f:**

R→ R, f(x) = [x], ∀ x ∈R> where [x] is the greatet integer less than or equal to x.**Square root function is f:**

R^{+}∪ {0} →R, f(x) = √x- Odd function. .

f(-x) = -f(x), ∀ x ∈D(f) **Even function.**

f(-x) = f(x), ∀ x ∈D (f)**Equal function,**two functions f and g are said to be equal if- D(f) = D(g)
- Co-domain of f = Co-domain of g
- f(x) = g(x), ∀ x ∈D (f), D(g)

**Addition**

(f+g) : X → R, (f + g)(x)

= f(x) + g(x), ∀ x ∈ X**Substraction**

(f – g): X → R, (f – g)(x)

= f(x) – g(x), ∀ x ∈ X**Scalar multiplication**

(kf): X →R,

(kf) (x) = k f(x),∀ x ∈ X and k∈ R**Multiplication (Product)**

(fg) : X → R,

(fg) (x) = f(x). g(x) , ∀ x ∈ X**Division (Quotient)**

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