## The Subject of Extending Logical Minds

Combine some logic with some bomb defusing; now how about understanding this to make your life easier?

• The original Logic manual can be found here.
• This contains ONLY specific logical identities and certain logic gate equivalents for this module and other "Logic" modules.
• The experts are allowed to use this for a better reference.

"A" and "B" are boolean variables in the reference sheet provided.

An "!" character represents a negate operator, E.G. !A represents NOT A.

Some Logic Identities
Involution !!A = A
Dominance (OR) TRUE ∨ A = TRUE
Dominance (AND) FALSE ∧ A = FALSE
Complementartity (OR) !A ∨ A = TRUE
Complementartity (AND) !A ∧ A = FALSE
DeMorgan's (OR) A ∨ B = !A | !B
DeMorgan's (AND) A ∧ B = !A ↓ !B
Absorption A ∧ (A ∨ B) = A
Idempotence A ∧ A = A ∨ A = A
Other Identities
A ↔ A = TRUE A ⊻ A = FALSE
Logical Gate Equivalents
A ↔ B = !A ⊻ B = A ⊻ !B = !(A ⊻ B) A ⊻ B = !A ↔ B = A ↔ !B = !(A ↔ B)
A → B = !A ∨ B A ← B = A ∨ !B
A ∨ B = A ← !B = !A → B A ↓ B = !(A ∨ B)
A | B = !(A ∧ B)

### Table ALT: Logical connective symbol list

Logical Connective Symbol Logic Gate Equivalent Interpterion
Conjunction AND Both TRUE returns TRUE. Otherwise at least 1 FALSE returns FALSE.
Disjunction OR At least 1 TRUE returns TRUE. Otherwise both FALSE returns FALSE.
Exclusive Disjunction XOR Matching inputs returns FALSE. NOT matching inputs returns TRUE.
Alternative Denial | NAND At least 1 FALSE returns TRUE. Otherwise, both TRUE returns FALSE.
Joint Denial NOR Both FALSE returns TRUE. Otherwise, at least 1 TRUE returns FALSE.
Biconditional XNOR Matching inputs returns TRUE. NOT matching inputs returns FALSE.
Implication (Left) - Left TRUE and right FALSE returns FALSE. Otherwise TRUE.
Implication (Right) - Left FALSE and right TRUE returns FALSE. Otherwise TRUE.