Download 4. Propositional Logic Using truth tables

Problems: Use the truth table method to
solve the following problems:
4. Propositional Logic
Using truth tables
1. Decide whether the formula p0→¬p0 is a
tautology, a contingency, or a contradiction.
2. Decide whether the formula p0v¬(p0∧p1) is a
tautology, a contingency, or a contradiction.
Problems
Jouko Väänänen: Propositional logic
Problems: Use the truth table method to
solve the following problems:
1. Decide whether p0→p1 is equivalent to
¬(p1→p0) or not.
2. Decide whether ¬p0 ∨p1 is equivalent to
¬(p0 ∧p1) or not.
Last
viewed
Problems: Use the truth table method to
prove the equivalence of
1. ¬(A ∧ B) and ¬A ∨ ¬B
2. ¬¬A and A
3. A∧A and A
4. A∨A and A
Jouko Väänänen: Propositional logic
Last
viewed
Problems: Use the truth table method to
prove the equivalence of
1. A→B and ¬A ∨ B
2. A!B and (A→B)∧(B→A)
Jouko Väänänen: Propositional logic
Last
viewed
Jouko Väänänen: Propositional logic
Last
viewed
Problems: Use the truth table method to
prove the equivalence of
1.
A∨(B∨C) and (A∨B)∨C
2.
A∧(B∧C) and (A∧B)∧C
Jouko Väänänen: Propositional logic
Last
viewed
Problems: Use the truth table method to
prove the equivalence of
Problems: Use the truth table method to
prove the equivalence of
1.
A∧(B∨C) and (A∧B)∨(A∧C)
1. A→(B→C) and B→(A→C)
2.
A∨(B∧C) and (A∨B)∧(A∨C)
2. A→(B→C) and (A∧B)→C
Jouko Väänänen: Propositional logic
Last
viewed
Exercise
!
Jouko Väänänen: Propositional logic
Last
viewed
Problem (Negation Normal Form)
Show that a conjunction of implications
pi→pi* is not a tautology if and only if
there is an i such that i is not i*.
Jouko "äänänen: Propositional logic
Define for formulas built up using just ¬, ∧ and ∨,
which are in negation normal form:
• (pi)*=pi, (¬pi)*=¬pi
(A∧B)*=A*∨B*
•
(A∨B)*=A*∧B*
Define for formulas built up using just ¬, ∧ and ∨:
!
(pi)+=pi, (pi)-=¬pi
!
(¬A)+=A-, (¬A)-=A+
(A∧B)+=A+∧B+, (A∧B)-=A-∨B-
!
(A∨B)+=A+∨B+, (A∨B)-=A-∧B-
!
A+ is called the Negation Normal Form of A. It has
negations only in front of proposition symbols.
!
!
Find (¬((p0∧p1)∨p2))+.
!
Show that A+ and A are equivalent.
!
Show that A- and ¬A are equivalent.
Last
viewed
Problem (Dual of a formula)
•
!
The formula A* is the dual of A. It is obtained
from A by switching conjunctions to disjunctions.
1. What is the dual of (p0∨¬p1)∧p2?
2. Show that if A is a tautology, so is ¬A*.
3. Show that if A and B are equivalent, then so
Jouko Väänänen: Propositional logic
Last
viewed
Problem (Substitution)
!
Suppose A is a propositional formula in
which the proposition symbols p0,…,pn-1
occur. Let Ai be for each i=0,…,n-1 an
arbitrary propositional formula. Let A’
be the result of substituting everywhere
in A the formula Ai for pi.
!
Show that if A is a tautology, then so is
A’.
are A* and B*.
Jouko Väänänen: Propositional logic
Last
viewed
Jouko Väänänen: Propositional logic
Last
viewed
Problem (Topological interpretation)
!
A topological interpretation is a topological space E and a
function f from proposition symbols to open sets in E. We
extend this to propositional formulas built up from ∧, ∨ and
¬:
"
f(A∧B)=f(A)⋂f(B)
"
f(A∨B)=f(A)∪f(B)
"
f(¬A)=Int(E-f(A)), the interior of the complement of f(A)
in the topological sense.
!
Show that A is a tautology iff f(A)=E in every topological
interpretation where E is a discrete space.
!
A is said to be a constructive tautology if f(A)=E in every
topological interpretation.
!
Show that ¬(A∧¬A) is, but A∨¬A is not a constructive
tautology.
Jouko Väänänen: Propositional logic
Last
viewed