A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.
→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).
⊃ may mean the same as ⇒ (the symbol may also mean superset).
A ⇔ B means A is true if B is true and A is false if B is false.
The statement ¬A is true if and only if A is false.
The statement A ∧ B is true if A and B are both true; else it is false.
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
∀ x: P(x) means P(x) is true for all x.
∃ x: P(x) means there is at least one x such that P(x) is true.
∃! x: P(x) means there is exactly one x such that P(x) is true.
x ⊢ y means y is derived from x.