If R and S are two symmetric relations then

Question

If R and S are two symmetric relations then (A) R o S is a symmetric relation (B) S o R is a symmetric relation (C) R ° S -1 is a symmetric relation (D) R o S is a symmetric relation if and only if R o S= SoR.

Tardigrade Admin · Accepted Answer

Since R and S are symmetric relations so R-1=R and S-1=S. But (R o S)-1=S-1 o R-1=S o R. Thus R o S is symmetric if and only if R o S=S o R.

Q. If $R$ and $S$ are two symmetric relations then

$R o S$ is a symmetric relation

$S o R$ is a symmetric relation

$R \circ S^{- 1}$ is a symmetric relation

$R o S$ is a symmetric relation if and only if $R o S =$ SoR.

Since $R$ and $S$ are symmetric relations so $R^{- 1} = R$ and $S^{- 1} = S$ . But $(R o S)^{- 1} = S^{- 1} o R^{- 1} = S o R$ . Thus $R o S$ is symmetric if and only if $R o S = S o R$ .

Q. If R and S are two symmetric relations then

RoS is a symmetric relation

SoR is a symmetric relation

R∘S−1 is a symmetric relation

RoS is a symmetric relation if and only if RoS= SoR.