element or subset

My task is consider the set

V = {b, d, f , g, {f , g}, {d, e, f} , {{d}, e} }

R = {c, d, e, f , g}

S = {f , g}

T = {d}

Classify each of the following statements as true or false.

(1) a ∈ V
(2) b ∈ V
(3) c ∉ V
(4) d ∉ V
(5) R ⊆ V
(6) S ⊆ V
(7) S ∈ V
(8) T ∈ V
(9) {d, e, f} ⊆ V
(10) {d, e, f} ∈ V
(11) {{d, e, f}} ⊆ V
(12) {{d, e, f}} ∈ V
(13) {T} ⊆ V
(14) {{T}} ⊆ V


1)  False
2)  True
3)  True
4)  False
5)  False
6)  False
7)  True
8)  False
9)  False
10) True
11) True
12) False
13) False
14) True


I`m especially concerned about the last three statements in which i am not really sure.

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(12)-(13) I agree with you, but about (14): it is true iff $\;\{T\}\in V\;$ ...is this last statement true? Since $\;T=\{d\}\;$, we have that $\;\{T\}=\{\{d\}\}\;$. Is this last an element of $\;V\;$ – DonAntonio Apr 19 '14 at 15:25

The only things that I could find is

• 6 is true

• 14 is false

For 6: The question is: Is $\{f,g \}\subset V$. This is true exactly when $f,g\in V$. This is true.

For 14: The questions is: Is $\{\{\{d\}\}\} \subset V$. This is true exactly when $\{\{d\}\} \in V$ and this is false.

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Would it be right to say for 6 that {f,g} ∈ V but {f,g} ⊈ V? – user3077612 Apr 19 '14 at 15:56
@user144257: Both of $\{f,g\}\in V$ and $\{f,g\}\subseteq V$ are true, but for different (and unrelated) reasons. $\{f,g\}\in V$ because $\{f,g\}$ is an element of $V$. And $\{f,g\}\subseteq V$ because $f$ is an element of $V$ and $g$ is an element of $V$. – Henning Makholm Apr 19 '14 at 16:02