Question

for each pair of statements, choose the one that is true.
(a)

• ( w in {w, x} )
• ( {w} in {w, x} )

(b)

• ( {1} subseteq {1, 2} )
• ( 1 subseteq {1, 2} )

(c)

• ( r subseteq {p, q, r} )
• ( r

otin {p, q, s} )
(d)

• ( {5, 6, 7} subseteq {1, 2, 3, 4, dots} )
• ( {5, 6, 7} in {1, 2, 3, 4, dots} )

Explanation:

Step1: Analyze set membership (a)

$\in$ denotes an element belongs to a set. $w$ is an element of $\{w, x\}$, while $\{w\}$ is a subset, not an element here.

Step2: Analyze subset relation (b)

$\subseteq$ means all elements of the first set are in the second. $\{1\}$'s element $1$ is in $\{1,2\}$; $1$ is an element, not a set, so $1 \subseteq \{1,2\}$ is invalid.

Step3: Analyze subset/element logic (c)

$r$ is an element, not a set, so $r \subseteq \{p,q,r\}$ is meaningless. $r
subseteq \{p,q,s\}$ is true because $r$ is not a set, so it cannot be a subset.

Step4: Analyze infinite set subset (d)

$\{1,2,3,4,...\}$ is the set of positive integers. $\{5,6,7\}$'s elements are all positive integers, so $\{5,6,7\} \subseteq \{1,2,3,4,...\}$ holds. $\{5,6,7\}$ is a set, not an element of the infinite set, so $\in$ is invalid.

Answer:

(a) $w \in \{w, x\}$
(b) $\{1\} \subseteq \{1, 2\}$
(c) $r
subseteq \{p, q, s\}$
(d) $\{5, 6, 7\} \subseteq \{1, 2, 3, 4, ...\}$