1. Logic
Is the study of reasoning; it is specifically concerned
with whether reasoning is correct.
It focuses on the relationship among statements as
opposed to the content of any particular statement.
Proposition
A sentence that is either true or false, but not both.
It is typically expressed as a declarative sentence (as
opposed to a question, command, etc.).
Propositions are the basic building blocks of any
theory of logic.
2. Which sentences are proposition?
1. The only positive integers that divide 7 are 1 and 7
itself.
2. Alfred Hitchcock won an Academy Award in 1940 for
directing “Rebecca.”
3. For every positive integer n, there is a prime number
larger than n.
4. Earth is the only planet in the universe that contains
life.
5. Buy two tickets to the “Never Say Never” concert.
3. We will use variables, such as p, q, and r, to represent
propositions.
Let p and q be propositions.
The conjunction of p and q, denoted p ʌ q, is the
proposition
p and q
The disjunction of p and q, denoted p v q, is the
proposition
p or q
4. example:
if
p: It is raining,
q: It is cold,
then the conjunction of p and q is
p ʌ q: It is raining and It is cold.
The disjunction of p and q is
p v q: It is raining or it is cold.
The inclusive-or of propositions p and q is true if both p
and q are true.
5. The exclusive-or that defines p exor q to be false if both
p and q are true.
Exclusive-or
p
T
T
F
F
q
T
F
T
F
p exor q
F
T
T
F
6. Truth tables
The truth values of propositions such as
conjunctions and disjunctions can be described by truth
tables.
Conjunction
Disjunction
pʌ
q
T
T
T
T
F
F
F
T
F
F
F
F
The binary operator on a set
elements in X an element of X.
p
q
p
T
T
F
F
q
T
F
T
F
pvq
T
T
T
F
X assigns to each pair of
ʌ and v are both binary operator on propositions.
7. The negation of p, denoted by ¬p, is the proposition
not p
example:
p: Paris is the capital of England.
¬p: Paris is not the capital of England.
Negation
p
T
F
¬p
F
T
A unary operator on a set X assigns to each element in
X an element of X.
¬ is a unary operator on propositions.
8. Operator Precedence
In expressions involving some or all of the
operators ¬, ʌ, and v, in the absence of parentheses, we
first evaluate ¬, then ʌ, and then v.
example: Given that proposition p is false, proposition
q is true, and proposition r is false, determine whether
the proposition
¬p v q ʌ r
is true or false.
9. Exercises:
I.Given that proposition p is false, proposition q is true,
and proposition r is false, determine whether each
proposition is true or false.
1. p v q
3. ¬p v q
5. ¬(p v q) ʌ (¬p v r)
II.
2. ¬p v ¬q
4. ¬p v ¬(q ʌ r)
6. (p v ¬r) ʌ ¬[(q v r) v ¬(r v p)]
Write the truth table of each proposition.
7. p ʌ ¬q
9. (p v q) ʌ ¬p
11. (p ʌ q) v (¬p v q)
13. ¬(p ʌ q) v (¬q v r)
8. (¬p v ¬q) v p
10. (p ʌ q) ʌ ¬p
12. ¬(p ʌ q) v (r ʌ ¬p)
14. (p ʌ q) ʌ ¬(r v p)
10. III. Represent the proposition symbolically by letting
p: There is a storm.
q: It is raining.
15. There is no storm.
16. There is a storm and it is raining.
17. There is a storm, and it is not raining.
18. There is no storm and it is not raining.
19. Either there is a storm or it is raining.
20. Either there is a storm or it is raining, but there is
no storm.
11. Conditionals
The implication (or conditional) p → q, is the
proposition “if p, then q”, that is false when p is true
and q is false and true otherwise. In this implication
p is called the hypothesis (or antecedent or
premise) and q is called the conclusion (or
consequence).
Implication
p
T
T
F
F
q
T
F
T
F
p →q
T
F
T
T
12. Other ways of expressing implication:
“p implies q”
“p is sufficient for q”
“when p then q”
“q if p”
“if p, q”
“q whenever p”
“p only if q”
“q is necessary for p”
Example: Rewrite the given implication in other
forms.
If it snows, then the streets are slick.
1.The streets are slick if it snows.
2.Snowy weather implies the streets are slick.
3.When it snows, then the streets are slick.
4.It snows only if the streets are slick.
13. Related Conditionals
The converse of the implication “if p, then q”, is the
implication “if q, then p.” That is, the converse of p→q
is q→p.
The inverse of the implication “if p, then q”, is the
implication “if not p, then not q” that results when p
and q are replaced by their negations; that is, the
inverse of p→q is ¬p→¬q
The contrapositive of the implication “if p, then q”, is
the implication “if not q, then not p.” That is, the
contrapositive of p→q is ¬q→¬p.
14. Example: Find the converse, inverse,
contrapositive of the given conditional.
and
If it rains, then I buy a new umbrella.
Converse:
If I buy a new umbrella, then it is raining.
Inverse:
If it does not rain, then I do not buy a new umbrella.
Contrapositive:
If I do not buy a new umbrella, then it is not raining.
15. The biconditional p↔q is the proposition that is
true when p and q have the same truth values and is
false otherwise.
“p if and only if q”
“p is necessary and sufficient for q”
“if p then q, and conversely”
Biconditional
p
T
T
F
F
q
T
F
T
F
p ↔q
T
F
F
T
16. Propositional Equivalences
Tautology is a compound proposition that is always
true, no matter what the truth values of the
propositions that occur in it.
Contradiction is a compound proposition that is
always false.
Contingency is a proposition that is neither a
tautology nor a contradiction.
Example of Tautology and Contradiction
p
T
F
¬p
F
T
p v ¬p
T
T
p ʌ ¬p
F
F
17. Logical Equivalence
Compound propositions that have the same truth
values in all positive cases are called logically
equivalent.
The propositions p and q are called logically
equivalent if p↔q is a tautology. The notation
p⇔q
denotes that p and q are logically equivalent.
Ex.
Show that the propositions p v (q ʌ r) and (p v q) ʌ
(p v r) are logically equivalent.
Table
18. Exercises:
I.Let A: “It is snowing”
B: “The roofs are white”
C: “The streets are slick”
D: “The trees are covered with ice”
Write the following in symbolic notation.
1. If it is snowing, then the trees are covered
with ice.
2. If it is not snowing, then the roofs are not
white.
3. If the streets are not slick, then it is not
snowing.
4. If the streets are slick, then the trees are not
covered with ice.
19. II. State the converse, inverse, and contrapositive of
each of the following conditionals.
1. If a triangle is a right triangle, then one angle has
a measure of 900.
2. If a number is a prime, then it is odd.
3. If two lines are parallel, then the alternate interior
angles are equal.
4. If Joyce is smiling, then she is happy.
III. Using truth tables, determine whether the
following pairs of statements are equivalent.
1. p v q; ¬p→q
2. ¬(p ʌ q); ¬p v ¬q
3. ¬(p v q); ¬p ʌ ¬q
4. p→q; ¬p→¬q
20. IV. Using truth tables, determine if the argument is
tautology, contradiction or contingency.
1.¬p → (p→q)
2.[p ʌ (p→q)] → ¬q
3.p ʌ ¬p
4.[(p→q) ʌ (q→r)] → (p→r)
5.[(p v q) ʌ (p→r) ʌ (q→r)] → r
6.(p→q) ʌ (p ʌ ¬q)
21. Quantifiers, Venn Diagrams, and Valid Arguments
Quantifiers give information about “how many” in the
statements in which they occur.
ex. all, some, no
Quantified Statements are statements involving
quantifiers.
ex. Some women have red hair.
All bananas are yellow.
No professors are bald-headed.
22. Universal Quantifiers ( ∀ )
The words all, every, and each are called universal
quantifiers because when these words are added to
an open sentence to make it a statement, the
sentence must be true in all possible instances in
order for the statement to be true.
∀xP(x)
“for all x P(x)” or “for every x P(x)
examples:
All prime numbers greater than 2 are odd.
Every automobile pollutes the atmosphere.
For each number x, x + 3 = 3 + x.
23. Existential Quantifiers ( ∃ )
Other quantified statements are intended to indicate
that there exists at least one case in which the
statement is true. Such statements generally involve
one of the existential quantifiers: some, there exist,
or there exists at least one.
∃xP(x)
“there is an x such that P(x)”
or “there is at least one x such that P(x)
examples:
Some men have black hair.
There exist students who do not work hard.
There exists at least one student who does not
work hard.
24. Venn Diagrams
A diagram in which the interiors of simple closed
curves such as circles are used to represent
collections of objects (or sets).
It provide geometrical representations of the
relationships indicated by statements involving
quantifiers.
ex. Draw Venn diagrams for the statements.
1.All dogs (D) are animals (A).
2.Some kindergarten students (K) are able to read
(R).
3.No cat (C) is a dog (D).
25. Valid Arguments
The argument is valid if in all cases where the
premises are true, the conclusion is true.
Ex. Consider the following two arguments. Use
Venn diagrams to determine whether either is valid.
a)All leghorns are chickens.
All chickens are fowls.
Therefore, all leghorns are fowls.
b)All leghorns are chickens.
All chickens are fowls.
Therefore, all fowls are leghorns.
26. If, in some case, the conjunction of the premises is
true and the conclusion is false, then the argument is
invalid. Invalid arguments are sometimes called
fallacies.
Theorem is a statement that can shown to be true.
Proof is a sequence of statements that form an
argument to demonstrate that a theorem is true.
Axioms or Postulates are the underlying
assumptions about mathematical structures, the
hypotheses of the theorem to be proved, and
previously proved theorem.
27. Lemma is a simple theorem used in the proof of
other theorems.
Corollary is a proposition that can be established
directly from a theorem that has been proved.
Conjecture is a statement whose truth value is
unknown. When a proof of a conjecture is found, the
conjecture becomes a theorem. Many times
conjectures are shown to be false, so they are not
theorems.
Rules of Inference which are means used to draw
conclusions from other assertions, tie together the
steps of a proof.
