Mixing logical notation with set theory notation

Question

Is it proper to mix logical notation with set theory notation? I would like to better understand when writing out "if" or "and" is necessary.

In these examples, the use of → to denote both a conditional and a mapping seems confusing. Is using a symbol to mean two different things a conflict?

"A equals B if every element x of A is an element of B and every element x of B is an element of A"

∀x[x ∈ A ⟹ x ∈ B] ∧ ∀x[x ∈ B ⟹ x ∈ A] ⟹ A = B

"A equals B if A and B have the same elements"?

∀x[x ∈ B ⟺ x ∈ A] ⟹ A = B

Greatly appreciated,

Answer 1

Indeed notations coincide, but often it is easy to recognize (from the context) which meaning you intended for the arrow ($\to$).

A mapping is an object from set theory, it is a mapping from a set to a set.

The arrow you use in the sentences are not arrows from sets to sets. They're arrows from statements about sets to statements about sets. So, they definitely are not functions that map sets to sets.

If you still want to avoid this, you can use use a regular arrow ($\to$) for mappings and double arrow ($\Rightarrow$) for implication between statements.

Answer 2

This the definition of set equality:

$$\forall A,B[A=B\iff (A\subseteq B\land B\subseteq A)]$$

Which translates to: for any two sets, they are equal if and only if they are subsets of one another (or, as you said they have exactly the same elements).

I hope that answers your question.

Answer 3

Many symbols in math are reused and need context to distinguish the possibilities. Rightward arrows can be mappings or can be implies or probably some other things I haven't thought of right now. In this case there are no mappings in sight, so I have no doubt the arrows are implies. If you ask for implies from MathJax you get $\implies$. If I had a problem that involved both mappings and implies, I would define different arrows in a preface.