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,
Best Answer
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.