Question

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

• ${v} in {p, v}$
• $v in {p, v}$

(b)

• ${11, 13, 15} in {1, 3, 5, 7, dots}$
• ${11, 13, 15} subseteq {1, 3, 5, 7, dots}$

(c)

• $z subseteq {t, z}$
• ${z} subseteq {t, z}$

(d)

• ${3}

subseteq {4, 5, 6}$

• ${3} in {3, 4, 5}$

Explanation:

Step1: Analyze set membership (a)

$\{v\}$ is a set, not an element of $\{p, v\}$; $v$ is an element, so $v \in \{p, v\}$ is true.

Step2: Analyze subset vs membership (b)

$\{11,13,15\}$ is a set of odd numbers, all elements are in $\{1,3,5,7,...\}$, so $\{11, 13, 15\} \subseteq \{1, 3, 5, 7, ...\}$ is true.

Step3: Analyze subset relation (c)

$z$ is not a set, so $z \subseteq \{t,z\}$ is invalid; $\{z\}$ is a set whose only element is in $\{t,z\}$, so $\{z\} \subseteq \{t, z\}$ is true.

Step4: Analyze non-subset (d)

$\{3\}$ is a set, not an element of $\{4,5,6\}$; $3$ is not in $\{4,5,6\}$, so $\{3\}
subseteq \{4, 5, 6\}$ is true.

Answer:

(a) $v \in \{p, v\}$
(b) $\{11, 13, 15\} \subseteq \{1, 3, 5, 7, ...\}$
(c) $\{z\} \subseteq \{t, z\}$
(d) $\{3\}
subseteq \{4, 5, 6\}$