Logic
Introduction
Discrete Propositional Mathematics
Andrei Bulatov
Discrete Mathematics Propositional Logic
Use of Logic
In mathematics and rhetoric:
Give precise meaning to statements.
Distinguish between valid and invalid arguments.
Provide rules of `correct reasoning.
Natural language can be very ambiguous
`If you do your homework, then youll get to watch the game.
`If you dont do your homework, then you will not get to watch ...
`You do your homework, or youll fail the exam.
=
`If you dont do your homework, then youll fail the exam.
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Discrete Mathematics Propositional Logic
Use of Logic (cntd)
In computing:
Derive new data / knowledge from existing facts
Design of computer circuits.
Construction of computer programs.
Verification of correctness of programs and circuit design.
Specification
What the customer
really needed
How the Programmer
understood it
What the customer
got
Discrete Mathematics Propositional Logic
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Statements (propositions)
Propositional logic deals with statements and their truth values
A statement is a declarative sentence that can be true or false
Truth values are TRUTH (T or 1) and FALSE (F or 0).
Examples:
- 1 + 1 = 2 (statement, T)
- The moon is made of cheese
(statement, F)
- Go home! (not statement, imperative)
- What a beautiful garden! (not statement, exclamation)
- Alice said, `What a beautiful garden! (statement, depends on Alice)
- y+1=2
- The God exists
(not statement, uncertain)
(statement, ?)
Discrete Mathematics Propositional Logic
Compound Statements
Simplest statement are called primitive statement
We cannot decide the truth value of a primitive statement. This is
not what logic does.
We shall use propositional variables to denote primitive
statements, p, q, r,
Instead we combine primitive statements by means of logic
connectives into compound statements or formulas and
look how the truth value of a compound statement depends on the
truth values of the primitive statements it includes.
We will denote compound statements by , ,
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Discrete Mathematics Propositional Logic
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Logic Connectives
negation (not, )
`It is not true that at least one politician was
honest
conjunction (and, ) `In this room there is a lady, and in the other
room there is a tiger
disjunction (or, )
`Margaret Mitchell wrote `Gone with the Wind
or I am going home
implication (if, then, )
`If there is a tiger in this room, then
there is no lady there
exclusive or (either , or , )
`There is either a tiger in this
room, or a lady
biconditional (equivalence) (if and only if, )
`There is a lady in this room if and only if
there is a tiger in the other room
Discrete Mathematics Propositional Logic
Truth tables
Truth table is a way to specify the exact dependence of the truth value
of a compound statement through the values of primitive statements
involved
truth values of primitive statements
(propositional variables)
truth value of compound statements
(formulas)
p
q
0
0
0
0
1
1
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Discrete Mathematics Propositional Logic
Truth Tables of Connectives (negation and conjunction)
p
p
F (0)
T (1)
T (1)
F (0)
Negation
Conjunction
unary connective
`Today is Friday
`Today is not Friday
p
q
pq
0
0
0
0
1
0
1
0
0
1
1
1
binary
connective
`Today is Friday
`It is raining
`Today is Friday and it
is raining
Discrete Mathematics Propositional Logic
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Truth Tables of Connectives (disjunction)
Disjunction `or
p
q
pq
0
0
0
0
1
1
1
0
1
1
1
`Students inclusive who have taken calculus
can take this course
`Students who have taken computing
can take this course
1
`Students who have taken calculus
or computing can take this course
Be careful with `or constructions in natural languages!
`You do your homework, or youll fail the exam.
`Today is Friday or Saturday
Discrete Mathematics Propositional Logic
Truth Tables of Connectives (exclusive or)
Exclusive `or
One of the statements is true but not both
p
q
pq
0
0
0
0
1
1
1
0
1
1
1
0
`You can follow the rules or be disqualified.
`Natalie will arrive today or Natalie will not arrive at all.
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Discrete Mathematics Propositional Logic
Truth Tables of Connectives (implication)
Implication
p
q
pq
0
0
1
0
1
1
1
0
0
1
1
1
Note that logical (material) implication does not assume any
causal connection.
`If black is white, then we live in Antarctic.
If pigs fly, then Paris is in France.
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Discrete Mathematics Propositional Logic
Implication as a promise
Implication can be thought of as a promise, and it is true if the
promise is kept
`If I am elected, then I will lower taxes
- He is not elected and taxes are not lowered
promise kept!
- He is not elected and taxes are lowered
promise kept!
- He is elected, but (=and) taxes are not lowered promise broken!
- He is elected and taxes are lowered
promise kept!
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Discrete Mathematics Propositional Logic
Playing with Implication
Parts of implication
pq
hypothesis
antecedent
premise
conclusion
consequence
`if p, then q
`if p, q
`p is sufficient for q
`q unless p
`p implies q
`p only if q
`q if p
`q when p
`q whenever p
`q follows from p
`a sufficient condition for q is p
`a necessary condition for p is q
Discrete Mathematics Propositional Logic
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Playing with Implication (cntd)
Converse, contrapositive, and inverse
pq
Converse
`The home team wins whenever it is raining
(`If it is raining then the home team wins)
qp
`If the home team wins, then it is raining
Contrapositive
q
p
`If the home team does not win, then it is not raining
Inverse
p
q
`If it is not raining, then the home team does not win
Discrete Mathematics Propositional Logic
Truth Tables of Connectives (biconditional)
Biconditional or Equivalence
One of the statements is true if and only if the other is true
p
q
pq
0
0
1
0
1
0
1
0
0
1
1
1
`You can take the flight if and only if you buy a ticket.
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Discrete Mathematics Propositional Logic
Example
`You can access the Internet from campus if you are a computer
science major or if you are not a freshman.
p - `you can access the Internet from campus
q - `you are a computer science major
r - `you are a freshman
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Discrete Mathematics Propositional Logic
Tautologies
Tautology is a compound statement (formula) that is true for all
combinations of truth values of its propositional variables
(p q) (q p)
p
q
(p q) (q p)
0
0
1
0
1
1
1
0
1
1
1
1
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Discrete Mathematics Propositional Logic
Contradictions
Contradiction is a compound statement (formula) that is false for
all combinations of truth values of its propositional variables
(p q) (p q)
p
q
(p q) (p q)
0
0
0
0
1
0
1
0
0
1
1
0
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Discrete Mathematics Propositional Logic
Homework
Exercises from the Book:
No. 1 ,3, 4, 8a, 8c (page 54)
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