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### Relation and Functions Class XII Chapter 1

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__Class 12 chapter 1__

__Relations and functions__

__Class 12 chapter 1__

__Relations and functions__

*Basic concepts of topic relations and functions class XII chapter 1 of mathematics. Equivalence relations, different types of functions, composition and inverse of functions.***Relations**

**Domain**

**Range**

**Types of relations**

**i) Empty Relation:**

**ii) Identity Relation**

**iii) Universal Relation :**

**iv) Reflexive relation**

**Number of reflexive relations**

^{5(5-1)}= 2

^{5x4}= 2

^{20}.

If n(A) = 3, the number of reflexive relations in a set A = 2

^{3(3-1)}= 2

^{3x2}= 2

^{6}= 64.

**v) Symmetric Relation**

**Number of Symmetric Relations**

**vi) Transitive Relation**

_{ }∈ A

**Number of Transitive Relations**

Number
of elements |
Number of Transitive relations |

0 |
1 |

1 |
2 |

2 |
13 |

3 |
171 |

4 |
3994 |

5 |
154303 |

6 |
9415189 |

**vii) Equivalence relation:**

**Number of equivalence relations**

Number of equivalence relations can be calculated as follows

For one element: 1

For two Elements: 1 2

For three Elements : 2 3 5

For four Elements : 5 7 10 15

For five Elements : 15 20 27 37 52

For six Elements : 52 67 87 114 151 203

Number
of elements |
No. of
equivalence relations |

1 |
1 |

2 |
2 |

3 |
5 |

4 |
15 |

5 |
52 |

6 |
203 |

**Viii) Equivalence Class:**

**An Empty relation R on the non-empty set S is not an equivalence relation**

**Explanation:**

**Reflexivity** fails because no element in set S is
related to itself (since the relation is empty, there are no pairs at all).

**Symmetry** and **Transitivity** are vacuously true
for the empty relation, because there are no pairs to contradict these
properties. However, reflexivity is a necessary condition for a relation to be
an equivalence relation, and since the empty relation does not satisfy
reflexivity, it cannot be considered an equivalence relation.

**An Empty relation R on the empty set S is an equivalence relation**

**Explanation**

**Reflexivity**

A relation R on a set S is reflexive if every element a ∈
S is related to itself, i.e., (a,
a)∈R
for all a ∈ S .

Since S is empty, there are no elements a ∈
S that need to satisfy (a, a)∈R. Hence, the reflexivity
condition is vacuously true.

**Symmetry**

A relation R on a set S is symmetric if for all a, b ∈ S,
whenever (a, b)∈R, it follows that (b, a) ∈ R.

Since S is empty, there are no pairs (a, b) to consider.
Hence, the symmetry condition is vacuously true.

**Transitivity**

A relation R on a set S is transitive if for all a, b, c ∈ S,
whenever (a, b)∈R and (b, c)∈R, it follows that (a, c)∈R.

Since S is empty, there are no pairs (a, b) and (b, c) to
consider. Hence, the transitivity condition is vacuously true.

Since the empty relation on the empty set vacuously
satisfies reflexivity, symmetry, and transitivity, it is indeed an equivalence
relation.

**Vacuously true :**

A statement is said to be "vacuously true" if it
is true in a trivial way due to the fact that there are no instances to
disprove it or if there is no counter example to disprove it.

**Trivial:**

In mathematics and logic, "trivial" typically
refers to something that is simple, straightforward, or self-evident. A trivial
case is often one that is so simple or obvious that it doesn't require much
proof or explanation.

**Trivial Relation**

A **trivial relation** in the context of set theory and
relations typically refers to one of two specific relations on a set:

**Empty Relation**: This is the relation where no element is related to any other element, including itself. Formally, for a set S, the empty relation R is defined as R = ∅. In other words, (a, b) ∉ R for any a, b ∈ S in S.**Universal Relation**: This is the relation where every element is related to every other element, including itself. Formally, for a set S, the universal relation R is defined as R=S×S. In other words, (a, b)∈R for all a, b ∈ S.

Examples of relations

(i) Symmetric but neither reflexive nor transitive.

Let relation is defined on A = {1, 2}

Answer: R = { (1, 1), (1, 2), (2, 1)}

(1, 2), (2, 1) ∈ R, so it is symmetric relation.

(2, 2) ∉ R so it is not reflexive.

(2,1) ∈ R and (1, 2) ∈ but (2, 2) ∉ R , so it is not transitive.

(ii) Transitive but neither reflexive nor symmetric.

Let a relation R is defined on a set A = {1, 2, 3}

Answer: {(1, 3)}

(1, 3) ∈ R and there is no ordered pair whose first element is 3, so it is a transitive relation.

(1, 1), (2, 2), (3, 3) ∉ R so it is not reflexive relation.

(1, 3) ∈ R but (3,1) ∉ R, so it is not symmetric relation.

(iii) Reflexive and symmetric but not transitive.

Let a relation R is defined on a set A = {1, 2, 3}

Answer: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3,1)}

(1, 1), (2, 2), (3, 3)∈ R so it is Reflexive.

(1, 2), (2, 1), (1, 3), (3, 1)∈ R so it is symmetric.

(3, 1)∈ R, and (1, 2) ∈ R but (3, 2) ∉ R so it is not transitive.

(iv) Reflexive and transitive but not symmetric.

Let a relation R is defined on a set A = {1, 2, 3}

Answer: R = {(1, 1), (2, 2), (1, 2)}

(1, 1), (2, 2) ∈ R, so it is a reflexive relation.

(1, 2) ∈ R but (2, 1) ∉ R so it is not symmetric.

(1, 2) ∈ R and there is no ordered pair whose first element is 2, so it is a transitive relation.

(v) Symmetric and transitive but not reflexive.

Answer: Let a relation R is defined on the set A = {1, 2, 3}

Ans: A = {(1, 1), (2, 2), (1, 2), (2, 1)}

(3, 3) ∉ R, so it is not reflexive relation.

(1, 2), (2, 1) ∈ R, so it is symmetric.

(1, 2) ∈ R, (2, 1) ∈ R, also (1,1) ∈ R so it is transitive relation.

^{n}

^{3 }= 64

^{4 }= 81

_{ }= b

Case 2 : For a, b ∈ A, such that : If a ≠ b ⇒ f(a) ≠ f(b)

^{n}P

_{n}= n!

^{4}P

_{3}= 4! = 24.

^{n}P

_{n}= n!

*******************************************

**Exercise 1.2**

_{f}= Odd natural number and Even natural numbers.

_{f}= All Natural Numbers = Co-domain

**Deleted Topics fro CBSE Syllabus**

^{-1}.

^{-1}exists.

^{-1}exists and we can write f

^{-1}(y) = x

^{-1}(1) = a, f

^{-1}(2) = b, f

^{-1}(3) = c, f

^{-1}(4) = d

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### Comments

Very nice and helpful thanks

ReplyDeleteThank u sir . Useful

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