Null set question

Here's something else about sets I'm trying to get right. The empty set is a set that contains nothing, written as $$\phi$$ = {}. It's called an empty set, so it is a set. Every set contains the empty set, right? Is there such a notion as an empty element? That doesn't sound right to me.

Normally we distinguish between an element and the set containing that single element, correct? But if the empty set is nothing (or the set that contains nothing) then the set {$$\phi$$} = {} = $$\phi$$. Is it proper to say that the empty set and it's power sets are the same? What would that make the cardinality of the empty set, simply zero?

tiny-tim
Homework Helper
Every set contains the empty set, right?

Hi Bob!

No … the power set of every set contains the empty set.

Hey Bob, this confuses me too sometimes here is how I think about it:

{$$\phi$$} = {}

This is not right. The RHS is the set that contains the empty set. The LRS is the empty set itself.

The empty set contains nothing. The set the contains the empty set contains something (the empty set)!

"Every set contains the empty set" is wrong. The correct statement is "every set contains the empty set as a subset".