Deciding if a statement is true or false regarding null sets/elements

[dirtypurp]

So far I have determined that first two statements are true but really need help on understanding proper logic as I don't think I am doing it right.

1 State whether the following are True or False

a.) {{φ}} ∈ {{φ}, {φ}}

b.) {φ} ∈ {{φ}, {φ}}

c.) {{φ}} ⊆ {φ, {φ}}

d.) φ ⊆ {φ, {φ}}

e.) φ ∈ {φ, {φ}}

f.) {φ} ⊆ {{φ, {φ}}}

 

[FactChecker]

You need to be careful about counting brackets. Make sure that a statement that refers to an element is in the other set as an element, not just a subset. For further help, you need to show how you are interpreting each side: element, set, subset, etc.

 

[member 587159]

So far I have determined that first two statements are true

First one is false.

 

[PeroK]

So far I have determined that first two statements are true but really need help on understanding proper logic as I don't think I am doing it right.

1 State whether the following are True or False

a.) {{φ}} ∈ {{φ}, {φ}}

b.) {φ} ∈ {{φ}, {φ}}

c.) {{φ}} ⊆ {φ, {φ}}

d.) φ ⊆ {φ, {φ}}

e.) φ ∈ {φ, {φ}}

f.) {φ} ⊆ {{φ, {φ}}}

What I would do first is replace all these sets with simpler notation. E.g:

$\varphi = A, \ \{ \varphi \} = \{ A \} = B, \ \{ \{ \varphi \} \} = \{ B \} = C$

 

[dirtypurp]

First one is false.

Yes I tried learning it again last night and I have gotten F, T, T, T, T, F.

 

[PeroK]

Yes I tried learning it again last night and I have gotten F, T, T, T, T, F.

I disagree with one of those answers.

 

[FactChecker]

Yes I tried learning it again last night and I have gotten F, T, T, T, T, F.

It's interesting that d and e force you to interpret the left side $\emptyset$ as a set and an element, respectively, to fit the context.

 

[member 587159]

It's interesting that d and e force you to interpret the left side $\emptyset$ as a set and an element, respectively, to fit the context.

Ah I didn't even make the connection that $\varphi$ was supposed to be empty set. Changes the solution of course.

 

[FactChecker]

Ah I didn't even make the connection that $\varphi$ was supposed to be empty set. Changes the solution of course.

That actually might be rust on my part. I guess it is not the symbol for empty set. Maybe it is not supposed to be the empty set.

 

[dirtypurp]

I disagree with one of those answers.

Can you clarify which one?

 

[PeroK]

Can you clarify which one?

Hmm, since they are true or false that's giving you the answer! Which one looks dodgy?

 

[dirtypurp]

Ah I didn't even make the connection that $\varphi$ was supposed to be empty set. Changes the solution of course.

So a empty set is a subset of any set making d true. For e since its asking if the empty set element can be found in {φ, {φ}} as the set does indeed contain a empty set. making it true as well.

 

[PeroK]

Ahh, so $\varphi$ is supposed to be the empty set! That changes things. Never seen it written like that. It should be $\emptyset$.

 

[dirtypurp]

Ahh, so $\varphi$ is supposed to be the empty set! That changes things. Never seen it written like that. It should be $\emptyset$.

With that being said - did i seem to get them all right from the previous response?

 

[PeroK]

With that being said - did i seem to get them all right from the previous response?

Yes, it was the confusion over using $\varphi$ for the empty set.

 

[Klystron]

The thread title uses yet a different term: null.

 

[zinq]

I believe all these text questions can be answered clearly without making any assumption about which set the symbol φ represents.

And just in case the profusion of suggestions has confused the original poster, it's safe to say that exactly half of the six statements are true and the other half are false.

 

[FactChecker]

I believe all these text questions can be answered clearly without making any assumption about which set the symbol φ represents.

In some, one must assume that it is a set and can be considered a subset. With that given, I think that I agree.

 

[zinq]

Yes, that's why I phrased it like that.