# Relations & Functions

**Cartesian Product—**

♦An ordered pair⟹(x,y)

♦An ordered triplet⟹(x,y,z)

♦Equality of ordered pairs ⟹ (x,y)=(α,β)⟺x=α,y=β

♦Equality of ordered triplets ⟹ (x,y,z)=(α,β,γ)⟺x=α, y=β, z=γ

Key Results: I.A (B C)=(A B)⋃(A C)

II.A (B⋂C)=(A B)⋂(A C)

III. A (B-C)=(A B)-(A C)

♦If A and B are two non-empty sets, then A B=B A⟺A=B

If A⊆B; then A A (A B)⋂(B A)

♦A⊆B⟹A C ⊆B C for any set C

♦A⊆B and C⊆D⟹A C ⊆ B D

♦(A B)⋃(C D)⊆(A C) (B⋃D)

♦(A B)⋂(C D)=(A⋂C) (B⋂D)

♦(A B)⋂(B A)=(A⋂B) (B⋂A)

♦Let A and B be two non-empty sets having n elements in common, then A B and B A have n^{2 }elements in common.

## Relations—

If A and B are any two non empty sets, then each subset of AxB is called a relation from A to B. It’s denoted by R:A⟶B

Key Point : I. If n(A)=p & n(B)=q, then the total number of relations from A to B =2^{pq }

♦ **Types of relations**:

**I.Empty Relation**: A relation having no element is called an empty relation.

Ex: R = {x : x+5=0 where x∊N}={}=ϕ

**II.Universal Relation**: If R:A⟶B is any relation such that R=AxB, then it’s called a universal relation.

Ex: A={1,2} & B={4,5}⟹AxB={(1,4),(1,5),(2,4),(2,5)}

R={(x,y):x+y<10 where x∊A, y∊B} = AxB

**III. Inverse Relation**: If R:A⟶B is any relation then its inverse relation R^{-1}:B⟶A is defined as follows: R^{-1} = {(y,x):(x,y)∊ R}

Ex: R={(1,2),(1,3),(2,2),(2,3)} ⟹ R^{-1}={(2,1),(3,1),(2,2),(3,2)}

**Functions— **

Name of the Functions |
Format |
Domain |
Range |

Constant Function | y=ƒ(x)=k, k∊R | R | {k} |

Identity Function⟹I | y=ƒ(x)=x | R | R |

Square Function | y=ƒ(x)=x^{2} |
R | [0,∞) |

Cube Function | y=ƒ(x)=x^{3} |
R | R |

Power Function | y=ƒ(x)=x^{n} |
R | R:when n is odd
[0,∞): when n is even |

Linear Function | y=ƒ(x)=ax+b, a≠0 | R | R |

Quadratic Function | y=ƒ(x)=ax^{2}+bx+c, a≠0 |
R | Calculate |

Polynomial Function | y=ƒ(x)=a_{0}x^{n}+a_{1}x^{n-1}+…a_{n},
where n,(n-1),…∊W & a |
R | Calculate |

Rational Function | y=ƒ(x)= ,h(x)≠0 | R-{x:h(x)=0} | Calculate |

Irrational Function | Having fractional powers of x | Calculate | Calculate |

Square Root Function | y=ƒ(x)= | [0,∞) | [0,∞) |

Cube Root Function | y=ƒ(x)= | R | R |

Signum Function | y=ƒ(x)= | R | {-1,0,1} |

Modulus Function | y=|x| | R | [0,∞) |

Exponential Function | y=a^{x },a>0,a≠1 |
R | (0,∞) |

Logarithmic Function | y=log_{a}x, a>0,a≠1,x>0 |
(0,∞) | R |

Greatest Integer Function | y=[x] | R | I |

Least Integer Function | y= | R | I |

Fractional Part Function | y={x} | R | [0,1) |

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