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Title: Discrete Mathematics
Description: Equivalent Propositions, Arguments, Rules of Inference
Description: Equivalent Propositions, Arguments, Rules of Inference
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Notes on Equivalent Propositions
• To show that two propositions (first proposition and
second proposition )are logically equivalent,
construct a truth table that establishes that the two
propositions have the same truth values for each of
the permutations of truth values to the component
propositions
...
SHOW THAT [P<=>Q] is equivalent to [(PQ) (P Q)]
P
Q
1
1
0
0
1
0
1
0
P<=>Q
1
0
0
1
PQ
1
0
0
0
PQ (PQ)(PQ)
0
0
0
1
1
0
0
1
• Corresponding TRUTH VALUES OF THE
PROPOSITIONS ARE EQUAL
...
SHOW THAT [P<=>Q] is equivalent to [(PQ) (P Q)]
i
...
SHOW THAT [P<=>Q] <=>[(PQ) (P Q)] is a tautology
PQ (PQ)(PQ) [P<=>Q] <=>[(PQ) (P Q)]
P
Q
P<=>Q
PQ
1
1
1
1
0
1
1
1
0
0
0
0
0
1
0
1
0
0
0
0
1
0
0
1
0
1
1
1
Since [P<=>Q] <=>[(PQ) (P Q)] is a tautology
...
SIMPLIFY [(PQ) v (PR)] ( Q v R )
[(P v Q )v(P v R)] (Q v R)
MI
[(P vP)v(Q v R)] ( Q v R )
Assoc
...
( Show that the left side (left of )
is the same as the right side (right of ))
[( QP ) ( P Q) (Q Q)] P
On a rare TV game show, a young couple was shown three keys labeled
A, B and C and was told that they could win a beautiful new home
simply by picking the key which opened its front door
...
“
• “ If key B will not open the door then key C will
...
”
After a few minute-reflection the couple made the correct choice
...
• “ Either key A will open the door, or key B will not
and key C will
...
“
• “ It is impossible that either both keys A and B will
open the door or both keys A and C will
...
“
•
“ If key B will not open the door then key C will
...
”
[(A
B
B) (A C)]
Let
A = Key A will open the door
B = Key B will open the door
C = Key C will open the door
C
Determine the permutation of truth values to the component
proposition where each of the 3 hints is true
...
[(A
B) (A C)]
0
0
0
1
1
1
1
1
Determine the permutation of truth values to the component
proposition where each of the 3 hints is true
...
(A( BC))( BC) ( [(AB)(AC)])
0
0
0
1
EXERCISE
• Prove each of the rules of replacement using
truth tables
Arguments
•
What is an argument?
An argument is a collection of propositions
where it is claimed that one of the propositions
(the conclusion ), follows from the other
propositions (the premises)
...
Deductive Argument
– Inductive Argument
One where it is claimed that within a certain
probability of error the conclusion follows from
the premises
– Deductive Argument
One where it is claimed that the conclusion
absolutely follows from the premises
...
Deductive Argument
– Inductive Argument
One where it is claimed that within a certain
probability of error the conclusion follows from
the premises
Inferential Statistics, Theory of Probability
– Deductive Argument
One where it is claimed that the conclusion
absolutely follows from the premises
...
Valid Argument
An Illustration
P
______
P v Q
Suppose P is true, PvQ is true
...
e
...
e
...
P
______
P v Q
Suppose P is true, PvQ is true
...
e
...
10 is
divisible by 2
...
P Q
P
______
Q
Suppose P Q and P are both true, Q must be true
...
Rules of Inference
(Valid Arguments)
• The rules of Inference are basic valid
arguments
...
• Illustrations
Rules of Inference
1
...
Conjunction
P
Q
______
PQ
Rules of Inference
3
...
Modus Ponens
P Q
P
______
Q
Rules of Inference
5
...
Disjunctive Syllogism
PvQ
P
______
Q
Rules of Inference
7
...
Constructive Dilemma
(PQ)(RS)
PvR
______
Q v S
Rules of Inference
9
Title: Discrete Mathematics
Description: Equivalent Propositions, Arguments, Rules of Inference
Description: Equivalent Propositions, Arguments, Rules of Inference