The universal set \(\varepsilon\) is the set of all integers and the subset P, Q, R of \(\varepsilon\) are given by: \(P = {x : x < 0} ; Q = {... , -5, -3, ...
Question 1 Report
The universal set \(\varepsilon\) is the set of all integers and the subset P, Q, R of \(\varepsilon\) are given by:
\(P = {x : x < 0} ; Q = {... , -5, -3, -1, 1, 3, 5} ; R = {x : -2 \leq x < 7}\)
(a) Find \(Q \cap R\).
(b) Find \(R'\) where R' is the complement of R with respect to \(\varepsilon\).
(c) Find \(P' \cup R'\)
(d) List the members of \((P \cap Q)'\).
Answer Details
The universal set \(\varepsilon\) is the set of all integers. Listing the given sets over the relevant range:
- \(P = \{x : x < 0\} = \{\dots, -3, -2, -1\}\) (negative integers)
- \(Q = \{\dots, -5, -3, -1, 1, 3, 5, \dots\}\) (odd integers)
- \(R = \{x : -2 \le x < 7\} = \{-2, -1, 0, 1, 2, 3, 4, 5, 6\}\)
(a) \(Q \cap R\) = odd integers lying in \(R\):
\[Q \cap R = \{-1,\, 1,\, 3,\, 5\}\]
(b) \(R'\) = all integers not in \(R\):
\[R' = \{\dots, -4, -3\} \cup \{7, 8, 9, \dots\} = \{x : x \le -3 \text{ or } x \ge 7\}\]
(c) \(P' = \{0, 1, 2, 3, \dots\}\) (non-negative integers). Then
\[P' \cup R' = \{0,1,2,\dots\} \cup \{\dots,-4,-3\}\cup\{7,8,\dots\} = \varepsilon \setminus \{-2,\,-1\}\]
i.e. every integer except \(-1\) and \(-2\).
(d) \(P \cap Q\) = negative odd integers \(= \{\dots, -5, -3, -1\}\). Its complement is every integer that is not a negative odd integer:
\[(P \cap Q)' = \{\dots, -6, -4, -2,\, 0, 1, 2, 3, 4, \dots\}\]
that is, all non-negative integers together with all negative even integers.
Download The App On Google Playstore
Everything you need to excel in your exams
