Question

If $R$ and $S$ are two non-void relations on a set $A$. Then, which of the following statements is incorrect $?$

• A. $R$ and $S$ are transitive implies $R\cap S$ is transitive
• B. $R$ and $S$ are transitive implies $R\cup S$ is transitive
• C. $R$ and $S$ are symmetric implies $R\cup S$ is symmetric
• D. $R$ and $S$ are reflexive implies $R\cap S$ is reflexive

Answer 1

The correct option is B

$R$ and $S$ are transitive implies $R\cup S$ is transitive

Checking for incorrect statement:

Step 1: Check for the condition if $R$ and $S$ are symmetric, then its union is symmetric or not.

A relation $R$ is reflexive if each element is related to itself.
A relation $R$ is symmetric if any one element is related to any other element, then the second element is related to the first.
A relation $R$ is transitive if any one element is related to a second and that second element is related to a third, then the first element is related to the third.

Let us consider two non-void relations on a set $A$ be:

$\begin{array}{l}R=\{(a,b),(b,a),(c,d),(d,c)\}\\ S=\{(x,y),(y,x),(e,f),(f,e)\}\end{array}$

Here, both $R$ and $S$ are symmetric, Then the union of $R$ and $S$ is given by

$R\cup S=\{(a,b),(b,a),(c,d),(d,c),(x,y),(y,x),(e,f),(f,e)\}$

Which implies that $R\cup S$ is symmetric.

Step 2: Check for the condition if $R$ and $S$ are reflexive, then its intersection is reflexive or not.

Let us consider two non-void relations on a set $A$ be:

$\begin{array}{l}R=\{(a,a),(b,b),(c,c),....,(n,n)\}\\ S=\{(x,x),(y,y),\left(a,a\right),...,(m,m)\}\end{array}$

Here, both $R$ and $S$ are reflexive, Then the union of $R$ and $S$ is given by

$R\cap S=(a,a)$ which implies that $R\cap S$ is reflexive.

Step 3: Check for the condition if $R$ and $S$ are transitive, then its union and intersection are transitive or not.

Let us consider two non-void relations on a set $A$ be:

$\begin{array}{l}R=\{(a,b),(b,c),(c,a),(e,d)\}\\ S=\{(x,y),(y,z),(z,x),(e,d)\}\end{array}$

Here both $R$ and $S$ are transitive, Then the union of $R$ and $S$ is given by

$R\cup S=\{(a,b),(b,c),(c,a),(e,d),(x,y),(y,z),(z,x)\}$

Then, $R\cup S$ is transitive.

Again check for the intersection of $R$ and $S$ is given by $R\cap S=\left(e,d\right)$,

Therefore,$R\cap S$ is not transitive.

Hence, option (B) is correct.