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For the set G where G = {{b},{c},d,{∅}}, I believe these are correct:

1) {{c}} [itex]\subseteq[/itex] G

2) {b} [itex]\subseteq[/itex] G

3) {d} [itex]\subseteq[/itex] G

4) d [itex]\subseteq[/itex] G

5) ∅ [itex]\subseteq[/itex] G

6) c [itex]\notin[/itex] [itex]\varphi[/itex](G)

7) {c} [itex]\in[/itex] [itex]\varphi[/itex](G)

8) {b} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

9) {{d}} [itex]\notin[/itex] [itex]\varphi[/itex](G)

10) ∅ [itex]\in[/itex] [itex]\varphi[/itex](G)

11) {∅} [itex]\in[/itex] [itex]\varphi[/itex](G)

12) ∅ [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

13) {∅} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

14) {{∅}} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

If any are incorrect an explanation would be appreciated.

1) {{c}} [itex]\subseteq[/itex] G

2) {b} [itex]\subseteq[/itex] G

3) {d} [itex]\subseteq[/itex] G

4) d [itex]\subseteq[/itex] G

5) ∅ [itex]\subseteq[/itex] G

6) c [itex]\notin[/itex] [itex]\varphi[/itex](G)

7) {c} [itex]\in[/itex] [itex]\varphi[/itex](G)

8) {b} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

9) {{d}} [itex]\notin[/itex] [itex]\varphi[/itex](G)

10) ∅ [itex]\in[/itex] [itex]\varphi[/itex](G)

11) {∅} [itex]\in[/itex] [itex]\varphi[/itex](G)

12) ∅ [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

13) {∅} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

14) {{∅}} [itex]\subseteq[/itex] [itex]\varphi[/itex](G)

If any are incorrect an explanation would be appreciated.

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