Question

State whether each of the following statements are true or false. If the statement is false rewrite the given statement correctly.
(i) If $P=\left\{ m,n \right\}$ and $Q=\left\{ n,m \right\}$, then $P\times Q=\left\{ \left( m,n \right),\left( n,m \right) \right\}$.
(ii) If A and B are non-empty sets then $A\times B$ is a non-empty set of ordered pairs $\left( x,y \right)$ such that $x\in A$ and $y\in B$.
(iii) If $A=\left\{ 1,2 \right\}$, $B=\left\{ 3,4 \right\}$ then $A\times \left( B\bigcap \phi \right)=\phi $

Answer 1

Hint: Here all given statements are related to set theory so we will use the various properties of sets to check whether the given statements are true or false. We will consider statements one by one and check, if any statement found false then we will rewrite the correct statement for it.

Complete step-by-step solution:
We have been given three statements.
We have to check whether each of the following statements is true or false.
Let us consider statements one by one and check.
(i) If $P=\left\{ m,n \right\}$ and $Q=\left\{ n,m \right\}$, then $P\times Q=\left\{ \left( m,n \right),\left( n,m \right) \right\}$.
In this statement we have been given two sets $P=\left\{ m,n \right\}$ and $Q=\left\{ n,m \right\}$.
The cross product of two sets is given as $P\times Q=\left\{ \left( m,n \right),\left( n,m \right) \right\}$.
Now, we know that the cross product of two sets is also called as the Cartesian product and is defined as- if set A has a number of elements and set B has b number of elements then the number of elements in the cross product will be $a\times b$.
So we get the cross product of $P\times Q$ as
$\Rightarrow P\times Q=\left\{ \left( m,n \right),\left( m,m \right),\left( n,m \right),\left( n,n \right) \right\}$
Hence the given statement is false.
The correct statement is if $P=\left\{ m,n \right\}$ and $Q=\left\{ n,m \right\}$, then $P\times Q=\left\{ \left( m,n \right),\left( m,m \right),\left( n,m \right),\left( n,n \right) \right\}$.
(ii) If A and B are non-empty sets then $A\times B$ is a non-empty set of ordered pairs $\left( x,y \right)$ such that $x\in A$ and $y\in B$.
Now, we know that the cross product of two non-empty sets A and B is the set of all possible ordered pairs $\left( x,y \right)$ such that $x\in A$ and $y\in B$ and it is denoted by $A\times B$.
Hence the given statement is true.
(iii) If $A=\left\{ 1,2 \right\}$, $B=\left\{ 3,4 \right\}$ then $A\times \left( B\bigcap \phi \right)=\phi $
As given in the statement A and B are non-empty sets and the cross product of sets will also be non-empty but the $\phi $ represents null sets.
Let us consider the LHS of the given equation then we will get
$\Rightarrow A\times \left( B\bigcap \phi \right)$
Now, we know that the intersection of an empty and non-empty set is a null set. Then substituting the value we will get
$\begin{align}
  & \Rightarrow A\times \phi \\
 & \Rightarrow \phi \\
\end{align}$
Now, we get LHS=RHS, hence the given statement is true.

Note: Here in this question $\phi $ represents the null set or empty set. We can represent the null set by using the symbol $\left\{ {} \right\}$. To solve these questions students must know the properties of sets and the cross product of sets.