Question

Given \[A = \{ x:x \in N{\text{ and }}3 < x \leqslant 6\} \] and \[B = \{ x:x \in W{\text{ and }}x < 4\} \]. Find the sets A and B in the roster form. Also, find \[A \cup B\] and \[A \cap B\].

Answer 1

Hint: The sets A and B are in set-builder form, convert them into roster form. Then, find \[A \cup B\] which contains the elements that belong to set A or B and \[A \cap B\] which contains elements that belong to both sets A and B.

Complete step-by-step answer:
A set is a collection of well-defined objects. It can be written in either set builder form or roster form.
In the set-builder form, the properties of the element of the set are written inside the braces and a variable is represented instead of the elements.
In roster form, all the elements of the set are listed inside the braces separated by commas.
We are given the set \[A = \{ x:x \in N{\text{ and }}3 < x \leqslant 6\} \], we express it in roster form.
\[A = \{ 4,5,6\} \]
Next, we have the set \[B = \{ x:x \in W{\text{ and }}x < 4\} \], we express it in roster form.
\[B = \{ 0,1,2,3\} \]
The number zero is included in set B because it is given that the elements belong to whole numbers.
Now, the set \[A \cup B\] contains the elements that belong to either set A or set B or both the sets. Hence, we have the following:
\[A \cup B = \{ 0,1,2,3,4,5,6\} \]
Next, the set \[A \cap B\] contains the elements that belong to both sets A and B. But we observe that there are no elements that are common to both the sets A and B.
\[A \cap B = \{ \} \]
\[A \cap B = \phi \]
The set \[\phi \] is an empty set.

Note: You may write the set B in roster form as {1, 2, 3} but it is wrong, the element 0 is missing. The number 0 also belongs to the whole numbers and the elements of set B are whole numbers less than 4.