hy Explain relations all its types with definition and function also.
Answers
Answer:
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Step-by-step explanation:
A function is a relation which describes that there should be only one output for each input. OR we can say that, a special kind of relation(a set of ordered pairs) which follows a rule i.e every X-value should be associated to only one y-value is called a Function.
For example:
Domain Range
-1 -3
1 3
3 9
the definition of Domain and Range of a function.
Domain:- It is a collection of the first values in the ordered pairs (Set of all input (x) values).
Range :-It is a collection of the second values in the ordered pairs (Set of all output (y) values).
Example:
In the relation, {(-2, 3), {4, 5), (6, -5), (-2, 3)},
The domain is {-2, 4, 6} and Range is {-5, 3, 5}.
Note: Don’t consider duplicates while writing Domain and Range and also write it in increasing order.
In terms of relations, we can define the types of functions as:
One to one function or Injective function: A function f: P → Q is said to be One to One if for each element of P there is a distinct element of Q.
Many to one function: A function which maps two or more elements of P to the same element of set Q.
Onto Function or Surjective function: A function for which every element of set Q there is pre-image in set P
One-one and Onto function or Bijective function: The function f matches with each element of P with a discrete element of Q and every element of Q has a pre-image in P.
Types of Relations
Different types of relations are as follows:
Empty Relations
Universal Relations
Identity Relations
Inverse Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Let us discuss all the types one by one.
Empty Relation
When there’s no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation also called as void. I.e R= ∅.
For example,
if there are 100 mangoes in the fruit basket. There’s no possibility of finding a relation R of getting any apple in the basket. So, R is Void as it has 100 mangoes and no apples.
Universal relation
R is a relation in a set, let’s say A is a universal Relation because, in this full relation, every element of A is related to every element of A. i.e R = A × A.
It’s a full relation as every element of Set A is in Set B.
Identity Relation
If every element of set A is related to itself only, it is called Identity relation.
I={(A, A), ∈ a}.
For Example,
When we throw a dice, the outcome we get is 36. I.e (1, 1) (1, 2), (1, 3)…..(6, 6). From these, if we consider the relation(1, 1), (2, 2), (3, 3) (4, 4) (5, 5) (6, 6), it is an identity relation.
Inverse Relation
If R is a relation from set A to set B i.e R ∈ A X B. The relation R−1 = {(b,a):(a,b) ∈ R}.
For Example,
If you throw two dice if R = {(1, 2) (2, 3)}, R−1= {(2, 1) (3, 2)}. Here the domain is the Range R−1 and vice versa.
Reflexive Relation
A relation is a reflexive relation If every element of set A maps to itself. I.e for every a ∈ A,(a, a) ∈ R.
Symmetric Relation
A symmetric relation is a relation R on a set A if (a,b) ∈ R then (b, a) ∈ R, for all a &b ∈ A.
Transitive Relation
If (a,b) ∈ R, (b,c) ∈ R, then (a,c) ∈ R, for all a,b,c ∈ A and this relation in set A is transitive.
Equivalence Relation
If and only if a relation is reflexive, symmetric and transitive, it is called an equivalence relation.
How to convert a Relation into a function?
A special kind of relation(a set of ordered pairs) which follows a rule i.e every X-value should be associated with only one y-value is called a Function.
Examples
Example 1: Is A = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)} a function?
Solution: If there are any duplicates or repetitions in the X-value, the relation is not a function.
But there’s a twist here. Look at the following example:
Relation Example
Though X-values are getting repeated here, still it is a function because they are associating with the same values of Y.
The point (1, 5) is repeated here twice and (3, -8) is written thrice. We can rewrite it by writing a single copy of the repeated ordered pairs. So, “A” is a function.
Example 2: Give an example of an Equivalence relation.
Solution:
If we note down all the outcomes of throwing two dice, it would include reflexive, symmetry and transitive relations. that will be called an Equivalence relation.
Example 3: All functions are relations, but not all relations are functions. Justify.
Solution:
Let’s suppose, we have two relations given in below table
A relation which is not a function A relation that is a function
Relation Function
As we can see duplication in X-values with different y-values, then this relation is not a function. As every value of X is different and is associated with only one value of y, this relation is a function
ᴀ ғᴜɴᴄᴛɪᴏɴ ɪs ᴀ ʀᴇʟᴀᴛɪᴏɴ ᴡʜɪᴄʜ ᴅᴇsᴄʀɪʙᴇs ᴛʜᴀᴛ ᴛʜᴇʀᴇ sʜᴏᴜʟᴅ ʙᴇ ᴏɴʟʏ ᴏɴᴇ ᴏᴜᴛᴘᴜᴛ ғᴏʀ ᴇᴀᴄʜ ɪɴᴘᴜᴛ. ᴏʀ ᴡᴇ ᴄᴀɴ sᴀʏ ᴛʜᴀᴛ, ᴀ sᴘᴇᴄɪᴀʟ ᴋɪɴᴅ ᴏғ ʀᴇʟᴀᴛɪᴏɴ(ᴀ sᴇᴛ ᴏғ ᴏʀᴅᴇʀᴇᴅ ᴘᴀɪʀs) ᴡʜɪᴄʜ ғᴏʟʟᴏᴡs ᴀ ʀᴜʟᴇ ɪ.ᴇ ᴇᴠᴇʀʏ x-ᴠᴀʟᴜᴇ sʜᴏᴜʟᴅ ʙᴇ ᴀssᴏᴄɪᴀᴛᴇᴅ ᴛᴏ ᴏɴʟʏ ᴏɴᴇ ʏ-ᴠᴀʟᴜᴇ ɪs ᴄᴀʟʟᴇᴅ ᴀ ғᴜɴᴄᴛɪᴏɴ. ғᴏʀ ᴇxᴀᴍᴘʟᴇ: ᴅᴏᴍᴀɪɴ