Question

for each pair of statements, choose the one that is true.
(a) $circ \\{r\\} \
subseteq \\{p, q, s\\}$
$\circ \\{r\\} \in \\{p, r, s\\}$
(b) $\circ \\{10, 12, 14\\} \subseteq \\{2, 4, 6, 8, \ldots\\}$
$\circ \\{10, 12, 14\\} \in \\{2, 4, 6, 8, \ldots\\}$
(c) $\circ \\{5\\} \in \\{5, 6\\}$
$\circ 5 \in \\{5, 6\\}$
(d) $\circ f \subseteq \\{f, g\\}$
$\circ \\{f\\} \subseteq \\{f, g\\}$

Explanation:

Step1: Analyze subset/membership (a)

$\{r\}$ is not a subset of $\{p,q,s\}$ (no $r$ in the set), and $\{r\}$ is not an element of $\{p,r,s\}$ (the set contains $r$, not $\{r\}$). So $\{r\}
subseteq \{p, q, s\}$ is true.

Step2: Analyze subset/membership (b)

$\{10,12,14\}$ elements are all even numbers, which are in $\{2,4,6,...\}$. $\subseteq$ means all elements belong, so $\{10, 12, 14\} \subseteq \{2, 4, 6, 8, ...\}$ is true.

Step3: Analyze subset/membership (c)

$\{5\}$ is a subset of $\{5,6\}$ (its element $5$ is in the set). $5 \in \{5,6\}$ is also true, but the option writes $5 \in \{5,6\}$ incorrectly as a set? No, correction: $\{5\} \subseteq \{5,6\}$ is valid subset notation, while $5 \in \{5,6\}$ is membership, but the option $\{5\} \subseteq \{5,6\}$ is the correct set-subset statement.

Step4: Analyze subset/membership (d)

$f$ is an element, not a set, so $f \subseteq \{f,g\}$ is invalid. $\{f\}$ is a set whose element is in $\{f,g\}$, so $\{f\} \subseteq \{f, g\}$ is true.

Answer:

(a) $\{r\}
subseteq \{p, q, s\}$
(b) $\{10, 12, 14\} \subseteq \{2, 4, 6, 8, ...\}$
(c) $\{5\} \subseteq \{5, 6\}$
(d) $\{f\} \subseteq \{f, g\}$