Question

Explain Antisymmetric relation with an example !!

┏(＾0＾)┛┏(＾0＾)┛​

Answer 1

Answer:

For example, 12 is divisible by 4, but 4 is not divisible by 12. The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

An example of antisymmetric is: for a relation “is divisible by” which is the relation for ordered pairs in the set of integers. For relation, R, an ordered pair (x,y) can be found where x and y are whole numbers and x is divisible by y.

In discrete Maths, a relation is said to be antisymmetric relation for a binary relation R on a set A, if there is no pair of distinct or dissimilar elements of A, each of which is related by R to the other. In a formal way, relation R is antisymmetric, specifically if for all a and b in A, if R(x, y) with x ≠ y, then R(y, x) must not hold, or, equivalently, if R(x, y) and R(y, x), then x = y. Hence, as per it, whenever (x,y) is in relation R, then (y, x) is not. Here x and y are the elements of set A. Apart from antisymmetric, there are different types of relations, such as:

●Reflexive

●Irreflexive

●Symmetric

●Asymmetric

●Transitive

An example of antisymmetric is: for a relation “is divisible by” which is the relation for ordered pairs in the set of integers. For relation, R, an ordered pair (x,y) can be found where x and y are whole numbers and x is divisible by y. It is not necessary that if a relation is antisymmetric then it holds R(x,x) for any value of x, which is the property of reflexive relation.

Antisymmetric Relation Definition

In set theory, the relation R is said to be antisymmetric on a set A, if xRy and yRx hold when x = y. Or it can be defined as, relation R is antisymmetric if either (x,y)∉R or (y,x)∉R whenever x ≠ y.

A relation R is not antisymmetric if there exist x,y∈A such that (x,y) ∈ R and (y,x) ∈ R but x ≠ y.

Note: If a relation is not symmetric that does not mean it is antisymmetric.

Antisymmetric Relation Examples

Q.1: Which of these are antisymmetric?

(i) R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)}

(ii) R = {(1,1),(1,3),(3,1)}

(iii) R = {(1,1),(1,2),(1,4),(2,1),(2,2),(3,3),(4,1),(4,4)}

Solution:

(i) R is not antisymmetric here because of (1,2) ∈ R and (2,1) ∈ R, but 1 ≠ 2.

(ii) R is not antisymmetric here because of (1,3) ∈ R and (3,1) ∈ R, but 1 ≠ 3.

(iii) R is not antisymmetric here because of (1,2) ∈ R and (2,1) ∈ R, but 1 ≠ 2 and also (1,4) ∈ R and (4,1) ∈ R but 1 ≠ 4.

STEP TO STEP EXPLANATION:

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Answer 2

ᴛᴇʀᴀ ᴀɴsᴡᴇʀ⛄ ::-

ᴀɴ ᴇxᴀᴍᴘʟᴇ ᴏғ ᴀɴᴛɪsʏᴍᴍᴇᴛʀɪᴄ ɪs: ғᴏʀ ᴀ ʀᴇʟᴀᴛɪᴏɴ “ɪs ᴅɪᴠɪsɪʙʟᴇ ʙʏ” ᴡʜɪᴄʜ ɪs ᴛʜᴇ ʀᴇʟᴀᴛɪᴏɴ ғᴏʀ ᴏʀᴅᴇʀᴇᴅ ᴘᴀɪʀs ɪɴ ᴛʜᴇ sᴇᴛ ᴏғ ɪɴᴛᴇɢᴇʀs. ғᴏʀ ʀᴇʟᴀᴛɪᴏɴ, ʀ, ᴀɴ ᴏʀᴅᴇʀᴇᴅ ᴘᴀɪʀ (x,ʏ) ᴄᴀɴ ʙᴇ ғᴏᴜɴᴅ ᴡʜᴇʀᴇ x ᴀɴᴅ ʏ ᴀʀᴇ ᴡʜᴏʟᴇ ɴᴜᴍʙᴇʀs ᴀɴᴅ x ɪs ᴅɪᴠɪsɪʙʟᴇ ʙʏ ʏ.

copied haii xD

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