 # If P = {a, b} and Q = {x, y, z}, show that P × Q ≠ Q × P.

Question:

If $P=\{a, b\}$ and $Q=\{x, y, z\}$, show that $P \times Q \neq Q \times P$.

Solution:

Given: P = {a, b} and Q = {x, y, z}

To show: $P \times Q \neq Q \times P$

Now, firstly we find the $P \times Q$ and $Q \times P$

By the definition of the Cartesian product,

Given two non - empty sets $P$ and $Q$. The Cartesian product $P \times Q$ is the set of all ordered pairs of elements from $\mathrm{P}$ and $\mathrm{Q}$, . i.e.

$P \times Q=\{(p, q): p \in P, q \in Q\}$

Here, $P=(a, b)$ and $Q=(x, y, z)$. So,

$P \times Q=(a, b) \times(x, y, z)$

$=\{(a, x),(a, y),(a, z),(b, x),(b, y),(b, z)\}$

$Q \times P=(x, y, z) \times(a, b)$

$=\{(x, a),(y, a),(z, a),(x, b),(y, b),(z, b)\}$

Since by the definition of equality of ordered pairs .i.e. the corresponding first elements are equal and the second elements are also equal, but here the pair (a, x) is not equal to the pair (x, a)

$\therefore \mathrm{P} \times \mathrm{Q} \neq \mathrm{Q} \times \mathrm{P}$

Hence proved