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