The universal set \(\varepsilon\) is the set of... | WAEC SSCE - General Mathematics - 1993 (Essay)

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 \(\varepsilon\).

(d) List the members of \((P \cap Q)'\).

Answer

The universal set \(\varepsilon\) is the set of all integers. Listing the given sets over the relevant range:

(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.

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