Question

which of the anti symmetric relation ?
R1
R2
R3
R4
all of these​

Answer 1

If $x\ R\ y$ and $y\ R\ x$ hold when $x = y$, the relation $R$ is said to be antisymmetric on a set $A$.

Step 1: Given data

$A=\{1,2,3,4\}$

Following relations on $A$-

$R1=\{(1,1),(2,2),(3,3)\}$

$R2=\{(1,2), (2,2), (2,3),(3,4), (4,1)\}$

$R3=\{(1,3),(2,2),(2,4),(3,1),(4,2)\}$

$R4=\{(1,3),(2,4),(3,1),(4,3)\}$

Antisymmetric relations$=?$

Step 2: Determining the antisymmetric relations

As per the given data,

• $R1$ is antisymmetric because $(1,1)\in R$ and $1=1$
• $R2$ is antisymmetric because $(2,2)\in R$ and $2=2$
• $R3$ is not antisymmetric because $(1,3)\in R$, $(3,1)\in R$ and $1\neq 3$
• $R4$ is not antisymmetric because $(1,3)\in R$, $(3,1)\in R$ and $1\neq 3$

Hence, $R1$ and $R2$ are antisymmetric.

#SPJ2