CSE390D Notes for Wednesday, 9/25/24
In lecture we discussed the idea that there are 16 possible truth tables for a
binary logical operator. Many of these 16 have a specific name or can be
described in a simple way. Some are somewhat stupid operators because they
ignore one or both of their arguments. Here are the tables we ended up
with:
| p |
q |
contradiction
F |
NOR
p ↓ q |
not reverse
implication? |
¬ p |
not
implication? |
¬ q |
XOR
p ⊕ q |
NAND
p | q |
| T | T |
F | F | F | F |
F | F | F | F |
| T | F |
F | F | F | F |
T | T | T | T |
| F | T |
F | F | T | T |
F | F | T | T |
| F | F |
F | T | F | T |
F | T | F | T |
| |
| p |
q |
and
p ∧ q |
biconditional
p ↔ q
also XNOR |
q |
implication
p → q |
p |
reverse
implication
p ← q |
or
p ∨ q |
tautology
T |
| T | T |
T | T | T | T |
T | T | T | T |
| T | F |
F | F | F | F |
T | T | T | T |
| F | T |
F | F | T | T |
F | F | T | T |
| F | F |
F | T | F | T |
F | T | F | T |
Some important vocabulary associated with these operators:
- proposition: a declarative sentence that is either true
or false
-
negation: the negation of a proposition p has the
opposite truth value and is often described as "not p"
-
conjunction, also known as logical and, p ∧ q is true
only if both p and q are true
-
disjunction, also known as logical or, p ∨ q is true if
either p or q is true
-
conditional, also known as implication, p → q is false
when p is true and q is false and true otherwise
-
converse, the converse of p → q is q → p
-
inverse, the inverse of p → q is ¬ p → ¬ q
-
contrapostive, the contrapositive of p → q is ¬ q → ¬ p
Stuart Reges
Last modified: Mon Sep 30 10:55:45 PDT 2024