You have enough to worry about memorizing without keeping dozens of symbols in your head at once. We intend to provide this table for your reference during every inclass evaluation.
true 
true 
True 
\top or 1 
1 
T, tautology 
false 
false 
False 
\bot or 0 
0 
F, contradiction 
not P 
!p 
not p 
\lnot P or \overline{P} 
~p 

P and Q 
p && q 
p and q 
P \land Q 
p & q 
P Q, P \cdot Q 
P or Q 
p  q 
p or q 
P \lor Q 
p  q 
P + Q 
P xor Q 
p != q 
p != q 
P \oplus Q 
p ^ q 
P ⊻ Q 
P implies Q 


P \rightarrow Q 

P \supset Q, P \Rightarrow Q 
P iff Q 
p == q 
p == q 
P \leftrightarrow Q 

P \Leftrightarrow Q, P xnor Q, P \equiv Q 
equivalent 
\equiv 
A \equiv B means A \leftrightarrow B is a tautology 
entails 
\vDash 
A \vDash B means A \rightarrow B is a tautology 
provable 
\vdash 
A \vdash B means both A \vDash B and I know B is true because A is true
\vdash B (i.e., without A) means I know B is true 
therefore 
\therefore 
\therefore A means both \vdash A and A is the thing we wanted to show 