Question

If \[A = \left\{ {a,b} \right\}\], \[B = \left\{ {c,d} \right\}\] and \[C = \left\{ {d,e} \right\}\]. Then find the set whose elements are \[\left\{ {\left( {a,c} \right), \left( {a,d} \right),\left( {a,e} \right),\left( {b,c} \right),\left( {b,d} \right),\left( {b,e} \right)} \right\}\]
A. \[A \cap \left( {BUC} \right)\]
B. \[AU\left( {B \cap C} \right)\]
C. \[A \times \left( {BUC} \right)\]
D. \[A \times \left( {B \cap C} \right)\]

Answer 1

Hint In the given question, the three sets and the set of ordered pairs are given. As it is a collection of ordered pairs, so there is a cross-product involved. We will identify the first term and second term of each ordered pair. Then we can easily recognize the set that is a collection of the first element and the second element of the ordered pairs. Then assign a cross-product between the sets.

Formula used
Set of ordered pairs: The pair of elements that occur in a particular order and are enclosed in a bracket is known as a set of ordered pairs.
If \[A\] and \[B\] are the two sets, then the cartesian product of the sets is: \[A \times B = \left\{ {\left( {a,b} \right):a \in A,b \in B} \right\}\]
The union of two sets is the set that contains all elements that are elements of both sets.

Complete step by step solution:
The given sets are \[A = \left\{ {a,b} \right\}\], \[B = \left\{ {c,d} \right\}\] and \[C = \left\{ {d,e} \right\}\]. And the elements of the set are \[\left\{ {\left( {a,c} \right), \left( {a,d} \right),\left( {a,e} \right),\left( {b,c} \right),\left( {b,d} \right),\left( {b,e} \right)} \right\}\].

The elements of the set \[\left\{ {\left( {a,c} \right), \left( {a,d} \right),\left( {a,e} \right),\left( {b,c} \right),\left( {b,d} \right),\left( {b,e} \right)} \right\}\] are the ordered pairs.
Since the ordered pairs are the cartesian product of two sets.
So, the set of first elements of ordered pairs in the given set is \[\left\{ {a,b} \right\}\] and the set of second elements is \[\left\{ {c,d,e} \right\}\].
Apply the definition of cartesian product of the sets.
\[\left\{ {\left( {a,c} \right), \left( {a,d} \right),\left( {a,e} \right),\left( {b,c} \right),\left( {b,d} \right),\left( {b,e} \right)} \right\} = \left\{ {a,b} \right\} \times \left\{ {c,d,e} \right\}\]
Now apply the definition of union of two sets.
\[\left\{ {\left( {a,c} \right), \left( {a,d} \right),\left( {a,e} \right),\left( {b,c} \right),\left( {b,d} \right),\left( {b,e} \right)} \right\} = A \times \left\{ {B \cup C} \right\}\] [Since \[c,d,e\] are the elements of the set \[B \cup C\] ]

Hence the correct option is option C.

Note: The cartesian product between two sets \[A\] and \[B\] is the all-possible ordered pairs with the first element from set \[A\] and the second element from set \[B\]. It is denoted by \[A \times B = \left\{ {\left( {a,b} \right):a \in A,b \in B} \right\}\]