Question

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

Answer 1

Here we will use the properties of sets to solve the given conditions one by one and verify whether they are true or false. Finally we conclude the required answer.

Complete step-by-step solution:
${\text{(i) if P = \\{ m, n\\} and Q = \\{ n, m\\} then P}} \times {\text{Q = \\{ (m, n),(n, m)\\} }}$
Now in the question the cross product between the sets $P$ and $Q$ is being done.
Now we can write the cross product of $P$ and $Q$as:
$P \times Q = \\{ m, n\\} \times \\{ n, m\\}$
On doing the cross product we get:
$P \times Q = \\{ (m, n),(m, m),(n, m),(n, n)\\}$
Now since the above given solution contradicts with the given cross product in the question which is:
${\text{P}} \times {\text{Q = \\{ (m, n),(n, m)\\} }}$
**We can conclude that the given statement is false.
The corrected version of the statement is: ${\text{if P = \\{ m, n\\} and Q = \\{ n, m\\} then P}} \times {\text{Q = \\{ (m, n),(m, m),(n, m),(n, n)\\} }}$ **
${\text{(ii)}}$ If ${\text{A}}$ and ${\text{B}}$ are non-empty sets then ${\text{A}} \times {\text{B}}$ is a non-empty set of ordered pairs ${\text{(x, y)}}$ such that ${\text{x}} \in {\text{A}}$ and ${\text{y}} \in {\text{B}}$
Since it is mentioned in the statement that the sets $A$ and $B$ are non-empty then the cross product of both the sets $A \times B$ will also be non-empty and will belong to the ordered pairs $(x, y)$ such that ${\text{x}} \in {\text{A}}$ and ${\text{y}} \in {\text{B}}$.
From the above statement $(i)$, we can see It is the property of cross product, that $A \times B$is a set of all the ordered pairs $(x, y)$ such that ${\text{x}} \in {\text{A}}$ and ${\text{y}} \in {\text{B}}$.
Therefore, the given statement is true.
${\text{(iii) if A = \\{ 1,2\\} ,B = \\{ 3,4\\} then A}} \times {\text{(B}} \cap \phi {\text{) = }}\phi$
We know that the sets $A$ and $B$ are non-empty sets since they have values in them. Now $\phi$ is null set.
Now the operation ${\text{A}} \times {\text{(B}} \cap \phi {\text{) = }}\phi$ can be written as:
Now on taking the left-hand side of the equation we get:
${\text{A}} \times {\text{(B}} \cap \phi {\text{)}}$
Since the intersection of an empty and non-empty set is a null set, it can be written as:
${\text{A}} \times \phi = \phi$
Similarly, the cross product will be null.
The given statement is true.

Note: The cross product of two sets is also called the Cartesian product of the sets and if there are two sets with elements $a$ and $b$ each, then the total number of elements in the cross product will be $a \times b$.
A null set which is represented using the symbol $\phi$ and $\\{ \\}$ is a set which has no terms in it. It is also called an empty set.