On the Subject of Boolean Venn Diagrams

Why is there a big Venn diagram? Why are there some weird symbols? Oh no...

See the original manual for the original instructions.

A     B     C A     B     C
X ∧ Y
AND
X ∨ Y
OR
X ⊻ Y
XOR
X → Y
IMPLIES
X | Y
NAND
X ↓ Y
NOR
X ↔ Y
XNOR
X ← Y
IMPLIED BY