Question

If $ A = \{ {a_1},{a_2}\} $ and $ B = \{ {b_1},{b_2},{b_3}\} $ then a subset of $ A \times B $ is:
a) $ \{ {a_1},{a_2}\} $
b) $ \{ {b_1},{b_2},{b_3}\} $
c) $ \{ ({a_1},{a_2}),({b_1},{b_2})\} $
d) $ \{ ({a_1},{b_1}),({a_1},{b_2})\} $

Answer 1

Hint: We are given two sets A and B and are asked to find that which of the options is a subset of $ A \times B $ . From the definition of $ A \times B $ it is a set containing elements of the form (x,y) where the first element is from the set A and the second element is from the set B or $ A \times B = \left\{ {(x,y)|x \in A,y \in B} \right\} $ . We will write $ A \times B $ in the roaster form and find out the option that contains elements of $ A \times B $ only.

Complete step-by-step answer:
From the definition above we have,
 $ A \times B = \left\{ {(x,y)|x \in A,y \in B} \right\} $
On listing the various elements and rewriting the set in roster form we have,
 $ \Rightarrow A \times B = \left\{ {({a_1},{b_1}),({a_1},{b_2}),({a_1},{b_3}),({a_2},{b_1}),({a_2},{b_2}),({a_2},{b_3})} \right\} $
Any subset of $ A \times B $ must not contain any element other than these
Clearly in option ‘a’ and ‘b’ we have
 $ \{ {a_1},{a_2}\} $ and $ \{ {b_1},{b_2},{b_3}\} $ respectively. Here as the set in option ‘a’ contains elements of set A only and option ‘b’ contains elements of set B only so these options can be discarded.
Now in option ‘c’ we have
 $ \Rightarrow \{ ({a_1},{a_2}),({b_1},{b_2})\} $ .
Clearly we don’t have any element $ ({a_1},{a_2}),({b_1},{b_2}) $ in $ A \times B $ that is available in this option so it can be discarded too.
Now we are left with the last option.
In option ‘d’ we have,
 $ \Rightarrow \{ ({a_1},{b_1}),({a_1},{b_2})\} $
Clearly this set is in the form of $ \left\{ {(x,y)|x \in A,y \in B} \right\} $ . Also the elements $ ({a_1},{b_1}),({a_1},{b_2}) $ are inside the set $ A \times B $ so option ‘d’ is correct.
So, the correct answer is “Option D”.

Note: Always list the elements of $ A \times B $ as it gives a clear picture about the subsets. We can easily check the options and discard the wrong ones and accept the correct ones. If the set $ A \times B $ is large and all elements cannot be listed then use the definition that $ A \times B = \left\{ {(x,y)|x \in A,y \in B} \right\} $ .