Discrrete mathematics for computer science propositional logic
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (146.45 KB, 10 trang )
Propositional Logic
03/22/19
1
Propositional Logic
• A proposition is a statement that is either true
or false.
• We give propositions names such as p, q, and r.
• E.g. p = “pigs have wings”
• q = “Obama is U.S. president”
• In this world anyway, p is false and q is true.
03/22/19
2
Truth Tables
“Not”
p
¬p
T
F
F
T
¬p is also written p
03/22/19
−
3
Logical
Operators/Connectives
• Building propositions from propositions
• “And”
p
q
p ∧q
T
T
T
T
F
F
F
T
F
F
F
F
• So “pigs have wings and Obama is US president” …
03/22/19
4
• “Or”
Logical
Operators/Connectives
p
q
p ∨q
T
T
T
T
F
T
F
T
T
F
F
F
• So “pigs have wings or Obama is US president” …
• “Or” is ambiguous in English but not in Propositional Logic
• E.g., “would you like coffee or ice cream?”
03/22/19
5
Logical
Operators/Connectives
“Exclusive or”
03/22/19
p
q
p⊕q
T
T
F
T
F
T
F
T
T
F
F
F
6
Some expressions can be
defined in terms of others
p ⊕ q is equivalent to (p∨q) ∧¬(p∧q)
What does this actually mean?
That the truth value of the formula on the left is
the same as the truth value of the formula on the
right regardless of the truth values of p and q
03/22/19
7
Propositional Logic
p
q
p⊕ q
pvq
p^q
¬(p∧q)
(p v q) ^
¬(p∧q)
T
T
F
T
T
F
F
T
F
T
T
F
T
T
F
T
T
T
F
T
T
F
F
F
F
F
T
F
03/22/19
8
Implies
p ⇒q or p→q or p⊃q
is equivalent to
¬p∨q
p
q
p⇒q
T
T
T
T
F
F
F
T
T
F
F
T
So “If pigs have wings then Obama is US president” is
true in this world
In fact a false proposition implies any proposition
03/22/19
9
“Iff” or logical equivalence
p
q
T
T
p⇔q
T
T
F
F
F
T
F
F
F
T
• p ⇔ q means that p and q have the same truth value
• Equivalent to (p ⇒ q) ∧ (q ⇒ p)
03/22/19
12
