Power of nothing!

Probability Level 3

What is true about the mathematical statements below?

$1)$ $\left\{ \emptyset \right\} \subset \left\{ \left\{ \emptyset \right\} \right\}$

$2)$ $\emptyset \subset \left\{ \left\{ \emptyset \right\} \right\}$

Details and Assumptions :

$\subset$ means is subset of.
$\emptyset$ has the same meaning as is generally used in set theory.

Only

2)

is correct. Both are correct. Both are wrong. Only

1)

is correct.

2 solutions

Venkata Karthik Bandaru
Jul 19, 2015

$\{\phi \} \subset \{ \{ \phi \} \} \Rightarrow \phi \in \{ \{ \phi \} \} \rightarrow \boxed{False}$ .

$\phi \subset A$ for every non-empty set A. Hence, $\phi \subset \{ \{ \phi \} \} \rightarrow \boxed{True}$ .

Therefore, only option $2)$ is correct.

Moderator note:

Great explanation.

The set of the empty set can be a very confusing idea.

$\subset$ is not the same as $\subseteq$ . None of these sets is a proper subset of the other.

Matthew Musick - 5 years, 10 months ago

$\phi$ is a proper subset of $\left\{ \left\{ \emptyset \right\} \right\}$

Venkata Karthik Bandaru - 5 years, 10 months ago

in 1st one, the RHS is a set of ------> a set of null sets. LHS is a single set of null set that belongs to that set of sets... described in RHS. a definite subset. in 2nd one, the RHS is still a set of ------->a set of null sets.. but LHS is a single null set not a subset.

Like {{1}} belongs to { {{1}}, {{2}}, ... } but {1} does not belong to { {{1}}, {{2}}, ... }

{1} wd belong to { {1}, {2,1,3}, ... } as it wd be the 1st element as u an see.

We need to remember, comma separates the elements of a set.

Please rectify ur ans.

Ananya Aaniya - 5 years, 10 months ago

I couldn't get you ma'm. Can you exactly pinpoint where the mistake is ?

Venkata Karthik Bandaru - 5 years, 10 months ago

"Phi" is NOT a member of {{"Phi"}} .--> since "Phi" is NOT a set of set..

{"Phi"} is a set of a Null Set. .. a valid member of {{"Phi"}}.

as par my understanding goes.

Ananya Aaniya - 5 years, 10 months ago

Rindell Mabunga
Jul 25, 2015

first is false since the greek symbol phi is not a subset of the set of the set that contains the greek symbol phi

second is true since null set is a subset of the set of the set that contains the greek symbol phi

Power of nothing!

2 solutions

Moderator note:

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