Definition of "unique" as it relates to sets?

Question

1) What is the definition of a "unique set?"

2) What is an example of a set that is not unique and how would you prove that it is not unique?

3) According to Wikipedia, "two sets are equal if and only if their 'membership requirements' are logically equivalent." Is it sufficient to show that a set is not unique if it is logically equivalent to another set, or is this considered to be different than uniqueness?

4) If so, I'm curious about this proof that the empty set is unique. If both E and E' are defined by the same "inclusion criteria" (i.e. that they are the empty set,) isn't it redundant afterwards to show that they are subsets of each other and thus equivalent?

Thank you

Answer 1

1) The word unique just means that there is only one set.

2) Say you had the following definition. Call a set an $5$-set if it has $5$ elements. Is there a unique $5$-set? I.e. does there exist only one set that is a $5$-set? Answer: No, because you have lots of technically different sets that contain five elements. Example $\{1,2,3,4,5\}$ and $\{6,7,8,9,10\}$.

3) I am not 100% sure that I understand this question. However, If you have a set $X$, and you define two subsets $Y=\{x\in X\lvert P(x)\}$ and $Z = \{x\in X\lvert W(x)\}$. Then the two sets are the same if and only if $P(x) \iff Q(x)$ (for $x\in X$). For example you could have $Y = \{x\in \mathbb{R}\lvert x\geq 0\}$, and $Z = \{x\in \mathbb{R}\lvert x-1 \geq -1\}$.

4) So you have a definition of an empty set and you here prove that there is only one (unique) set that satisfies this definition.

Answer 2

Unique means there is only one such set. For example $\emptyset$ is unique because if we suppose there are two such sets, we will show that they are subsets of each other, meaning it is the same set. Nothing in set theory (axiomatic that is) is redundant every axiom must be precise and necessary. For example to use equality we need the axiom of extensionality. It might seem unnecessary but otherwise we have no notion of equality of sets. Uniqueness makes more sense when we use sets in some context, for example you might know from linear algebra that a vector space does NOT have a unique basis, which is a set of linearly independent vectors.