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:

Part (a)

Step1: Recall set membership and element vs set

The symbol $\in$ means "is an element of". A set like $\{v\}$ is a set containing the element $v$, while $v$ is an element. The set $\{p, v\}$ has elements $p$ and $v$. So $v$ is an element of $\{p, v\}$, but $\{v\}$ is a set, not an element of $\{p, v\}$.

Step2: Conclusion for (a)

So $v \in \{p, v\}$ is true.

Part (b)

Step1: Recall $\in$ and $\subseteq$

$\in$ is for element - set, $\subseteq$ is for set - set where every element of the first set is in the second. The set $\{1, 3, 5, 7, \dots\}$ is the set of odd positive integers. $\{11, 13, 15\}$ is a set of odd integers. Every element of $\{11, 13, 15\}$ is in $\{1, 3, 5, 7, \dots\}$. But $\{11, 13, 15\}$ is not an element of $\{1, 3, 5, 7, \dots\}$ (the elements of $\{1, 3, 5, 7, \dots\}$ are numbers like 1, 3, 5, etc., not sets like $\{11, 13, 15\}$).

Step2: Conclusion for (b)

So $\{11, 13, 15\} \subseteq \{1, 3, 5, 7, \dots\}$ is true.

Part (c)

Step1: Recall element vs subset

The symbol $\subseteq$ means "is a subset of". An element $z$ cannot be a subset (unless we are in a very specific context, but generally, subsets are sets). The set $\{z\}$ is a set containing $z$. For a set $A$ to be a subset of $B$, every element of $A$ must be in $B$. The set $\{z\}$ has element $z$, and $z$ is in $\{t, z\}$. So $\{z\} \subseteq \{t, z\}$ is true, while $z$ is an element, not a subset.

Step2: Conclusion for (c)

So $\{z\} \subseteq \{t, z\}$ is true.

Part (d)

Answer:

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