2. <<Title>>
Acknowledgement
These slides have been
adapted from:
Epp, S.S., 2010. Discrete mathematics
with applications. Cengage learning.
Hammack, R.H., 2013. Book of proof.
Richard Hammack.
3. LO 1 :
Explain basics concepts of logic,
methods of proof, set theory and
function.
.
6. LOGIC
Logic is a systematic way of thinking
that allow us to deduce new
information from old information and
to parse the meaning of sentences
7. Not Proposition Proposition
Add 5 to both sides!.
What is the solution
of 2x = 8 ?
Adding 5 to both
sides of x-5=37 gives
x=42.
The solution of
2x = 8 is 42
A Statement (or
Proposition) is a
sentence that is
true or false but
not both
8. Negation Conjunction Disjunction
Definition
Truth Table
Definition Definition
If p is a proposition
variable, the
negation of p is “not
p” or “It is not the
case that
p” and is denoted
∼p.
If p and q are
proposition variables,
the conjunction of p
and q is “p and q,”
denoted
p q.
Truth Table Truth Table
If p and q are
statement variables,
the disjunction of p
and q is “p or q,”
denoted p q
9. Example
P : The number 2 is even
Q : The number 3 is odd
Negation
~P: The number 2 is not even;
~P: It is false that the number 2 even;
Conjunction
P Q : The number 2 is even and the number 3 is od
Disjunction
P Q : The number 2 is even or the number 3 is od
10. Construct the truth table for the
statement form (p q) ∼r.
Construct the truth table for the
statement form (p v q) ∼(p q).
11. Two proposition forms are called logically equivalent if, and only if, they have
identical truth values for each possible substitution of statements for their
statement variables. The logical equivalence of statement forms P and Q is
denoted by writing P ≡ Q. There are two technique to proof equivalence from
two proposition such as by using truth table and proof by using theorem.
Logical Equivalence
12. Logical Equivalence
Example 1 Example 2
From Example 1 and Example 2, how to proof equivalence two proposition by
using a truth table
13. A tautology is a statement
form that is always true.
A statement whose form is
a tautology is a tautological
statement.
Tautology and Contradiction
A contradiction is a statement form
that is always false.
A statement whose form is a
contradiction is a contradictory
statement.
Example : Show that the statement
form p ∼p is a tautology and that
the statement form p ∼p is a
contradiction.
14. Theorem of Logical Equivalences
Given any proposition variables p, q, and r . a tautology t and a contradiction c, the
following logical equivalences hold.
Summary of Logical Equivalences
15. Example: Use Theorem of Logical Equivalences to verify the logical equivalence
∼(∼p ∧ q) ∧ (p ∨ q) ≡ p.
Solution
The second technique to proof equivalence two proposition is by
using Theorem of Logical Equivalences . From an example above,
we see how to use it.
17. Conditional Statements Biconditional Statements
If p and q are proposition variables,
the conditional of q by p is “If p
then q” or “p implies q” and is
denoted
p → q.
- p the hypothesis of the conditional.
- q the conclusion.
Given statement variables p and q,
the biconditional of p and q is “p if,
and only if, q” and is denoted
p ↔ q.
18. The negation of “if p then q” is logically equivalent to “p and not q.”
The contrapositive of “if p then q” is “If ~q then ~p”
The converse of “if p then q” is “If q then
p“
The inverse of “if p then q” is “If ∼p then ∼q.”
~(p→q ) ≡ p ~q
The contrapositive of p→q is
~q→~p
The converse of p→q is (q → p)
The inverse of p→q is (~p → ~q)
19. The negation : Howard can swim across the lake and Howard cannot swim
to the island.
The contrapositive : If Howard cannot swim to the island, then Howard cannot swim
across the lake.
Example : If Howard can swim across the lake, then Howard can swim to the
island.
Write the,negation, contrapositive, converse, and inverse from the statements:
The converse : If Howard can swim to the island, then Howard can swim across
the lake.
The inverse : If Howard cannot swim across the lake, then Howard cannot swim to
the island.
20.
21. An argument is a sequence of statements (proposition).
Argument
Valid
To say that an argument is valid means that if the
resulting premises are all true, then the conclusion is
also true.
All statements (or proposition) in an argument except for the final one
, are called premises (or assumptions or hypotheses).
The final statement or statement form is called the conclusion.
The symbol ∴ , which is read "therefore," is normally placed just before
the conclusion.
24. The operation of a black box is completely specified by
constructing an input/output table that lists all its possible input
signals together with their corresponding output signals.
Black Boxes and Gates
29. A predicate is a
sentence that contains
a finite number of
variables and becomes
a propositions when
specific values are
substituted for the
variables.
Let P(x) be the predicate "x2 > x" with domain the
set R of all real numbers. Write P(2), P(1/2), P(-2),
and indicate which of these propositions
are true and which are false!
P(2) : 22 > 2 or 4 > 2. TRUE
P(1/2) : (1/2) 2 > (1/2) or (1/4) > (1/2). FALSE
P(-2) : (-2)2 > 2 or 4 > 2. TRUE
Example
30. The Universal Quantifier,
The symbol denotes "for all" and is
called the universal quantifier.
Example :
The sentence "All human beings are
mortal" is to write
human beings x, x is mortal
or, more formally
x S, x is mortal,
where S denotes the set of all human
beings.
The Existential Quantifier,
The symbol denotes "there exists"
and is called the existential quantifier.
Example:
The sentence "There is a student in
Math 140" can be written as
a person s such that s is a student in
Math 140,
or more formally,
s S such that s is a student in Math
140,
where S is the set of all people.
31. Truth and Falsity of Universal
Proposition
Let Q(x) be a predicate and D the
domain of x. A universal proposition
is a proposition of the form "x D,
Q(x)“.
It is defined to be true if, and only if,
Q(x) is true for every x in D.
It is defined to be false if, and only if,
Q(x) is false for at least one x in D.
A value for x for which Q (x) is false is
called a counter example to the
universal proposition.
Truth and Falsity Existential Proposition
The symbol denotes "there exists"
and is called the existential quantifier.
Example:
The sentence "There is a student in
Math 140" can be written as
a person s such that s is a student in
Math 140,
or more formally,
s S such that s is a student in Math
140,
where S is the set of all people.
32. Epp, S.Susanna., 2010. Discrete mathematics with applications.
Cengage learning.
Hammack, R.H., 2013. Book of proof. Richard Hammack.
REFERENCES
