Carlos Aguilar
DISCRETE MATHEMATICS
Jose De Jesus
Melendez
Angie Rangel
Drake
Mackenzie
Jain
SECTIONS
 Section 2.4:
Application: Digital Logic Circuits
# 34
 Section 3.1:
Predicates and Quantified Statements I
# 16, 21
 Section 3.2:
Predicates and Quantified Statements II
#14, 38
TERMS/SYMBOLS
 Conjunction (Symbol: ˄) “ p ˄ q ” = p conjunction q (p and q) is
a new true statement if both p and q are true statements .
 Disjunction (Symbol: ˅) “ p ˅ q ” = p disjunction q (p or q) is a
new false statement if both p and q are false statements .
 Conditional Statement (Symbol :→) “ p → q ” = (if p, then q) a
new statement that is false when p is a true statement and q
is a false statement.
TERMS/SYMBOLS
 Biconditional Statement (Symbol: ↔) “ p ↔ q ” (p if and only if
q) = a new true statement if both p and q have same truth
values.
 Negation Statement (Symbol: ~) = “ ~p ” (not p) = a statement
in which the new statement is opposite from the original
statement.
SECTION 2.4 APPLICATION: DIGITAL LOGIC CIRCUITS
NOR Operator
Peirce Arrow ( symbol: ↓ )
NOR-gate is a gate that acts like an OR-gate, which is
then followed by a NOT-gate. When both input signals are
0, then the output signal for a NOR -gate is 1 .
 Notation:
P↓Q
(meaning neither P nor Q)
≡ ~ (P ˅ Q)
SHOW THAT THE FOLLOWING LOGICAL
EQUIVALENCES HOLD FOR THE PEIRCE
ARROW ↓, WHERE P↓Q ≡~(P˅Q).
 a.
~P ≡ P ↓ P
 b.
P ˅ Q ≡ (P ↓ Q) ↓ (P ↓ Q)
 c.
P ˄ Q ≡ (P ↓ P) ↓ (Q ↓ Q)
 d.
Write P → Q using Peirce arrows only.
 e.
Write P ↔ Q using Peirce arrows only.
SOLUTIONS
a.
~P ≡ P ↓ P
~P ≡ ~ (P ˅ P)
≡P↓P
b.
by the idempotent law for ˅
by definition of ↓
P ˅ Q ≡ (P ↓ Q) ↓ (P ↓ Q)
≡ ~(P ↓ Q)
≡ ~(~P ˄ ~Q)
≡P˅Q
by definition of ↓
by the double negation law
c.
P
≡
≡
≡
P ˄ Q ≡ (P ↓ P) ↓ (Q ↓ Q)
˄ Q ≡ ~(~(P ˄ Q))
~(~P ˅ ~Q)
~((P ↓ P) ˅ (Q ↓ Q))
(P ↓ P) ↓ (Q ↓ Q)
by the double negation law
by De Morgan’s law
by definition of ↓
d. Write P → Q using Peirce arrows only.
P
≡
≡
≡
≡
≡
→ Q ≡ (~P ↓ Q)
~ (~P ↓ Q)
~ (~ (~P ˅ Q))
~ (P ˄ ~Q)
~P ˅ Q
P→Q
↓ (~P ↓ Q)
by definition of ↓
by the double negation law
by De Morgan’s Law
e. Write P ↔ Q using Peirce arrows only.
P ↔ Q ≡ (~P ↓ Q) ↓ (P ↓ ~Q)
≡ ~ (~P ˅ Q) ↓ ~ (P ˅ ~Q)
by definition of ↓
≡ (P ˄ ~Q) ↓ (~P ˄ Q)
by De Morgan’s Law
≡ ~ ((P ˄ ~Q) ˅ (~P ˄ Q))
by definition of ↓ and De Morgan’s
Law
≡ (~P ˅ Q) ˄ (P ˅ ~Q)
≡P↔Q
UNIVERSAL
QUANTIFIERS
(DISCRETE MATHEMATICS SECTION 3.1)
By: Jose Melendez
BEFORE THAT
Predicates
&
Quantifiers
&
Charles S. Pierce
THE MAN BEHIND THE LOGIC
 Charles Sanders Pierce
 1839 – 1914
 Sometimes known as “Father of
Pragmatism”
 B.A., B.Sc., M.A. from Harvard University
 Mathematician, astronomer, chemist,
engineer, dramatist, etc. (Brent 2)
 “[I intend] to make a philosophy like
that of Aristotle… and in whatever other
department there may be, shall appear
as the filling up of its details.” – Charles
Pierce
PREDICATES
Definition of predicates:
- A sentence that contains a finite number of
variables and becomes a statement when specific
values are substituted for the variables “x” (Epp
97)
- The domain of a predicate variable is the set of all
values that may be substituted in place of the
variable “x” (Epp 97)
P(x)

Used to represent predicates
QUANTIFIERS
Definition of quantifiers:
- Words that refer to quantities such as “some” or
“all” and tell for how many elements a given
predicate is true (Epp 97)


Universal
(For all)


Existential
(There Exists)
FOCUS: UNIVERSAL QUANTIFIERS
“”
 Stands “for all”
 Statement written as “ _____ x, x _____”, where the
blank between “” and “x” is considered the main
point of the sentence; the “x _____” refers to as the
domain
 Domain and main point should be put into singular
form
HOW IT IS DONE
 All soccer balls are round.
1. Main point?
- soccer balls
2. Domain?
- round
3. Statement?
-  soccer ball x, x is round
HOW IT IS DONE
 The number 3 is not equal to the square of any real
number.
1. Main point?
- real number (why: keyword “any” refers to
real number)
2. Domain?
- 𝑥 2 is not equal to 3
3. Statement?
-  real number x, 𝑥 2 is not equal to 3
EXAMPLES
 Rewrite each of the following statements in the form of
“ _______ x, _______.”
 1. All dinosaurs are extinct.
 dinosaur x, x is extinct.
 2. Every real number is positive, negative, or zero.
 Real number x, x is positive, negative, or zero.
 3. No irrational numbers are integers.
 irrational number x, x is not an integer
 4. No logicians are lazy.
 logician x, x is not lazy
MORE EXAMPLES
 5. The number 2,147,581,953 is not equal to the square of any
integer.
 integer x, 𝑥 2 is not equal to the square of any integer
 6. The number -1 is not equal to the square of any real number.
 real number x, 𝑥 2 is not equal to -1
UNIVERSAL STATEMENT: ( ∀ )
A Universal statement includes the quantifier
( ∀ ) = “all”
For Example:
∀x, if x is chocolate, then x tastes good
UNIVERSAL EXAMPLE
∀x, if x is tastes good, then x is chocolate.
Negation is….
∃x, such that x tastes bad then x is not chocolate.
EXISTENTIAL QUANTIFIER
AND FORMAL STATEMENT : ( ∃ )
 A Existential statement includes the quantifier
( ∃ ) = “some”
Formal Statement: ∃b a gamer, such that b is boy.
EXISTENTIAL EXAMPLE
∃b a boy, such that b is gamer.
Negation is…
∀b is a boy, if b is a gamer then b is a boy.
INFORMAL LANGUAGE OF LOGIC:
 The words such that are inserted just before the
predicate. Some other expressions that can be used in
place of there exists are there is a, we can find a, there
is at least one, for some, and for at least one.
PREDICATES AND QUANTIFIED
STATEMENTS I
(a.) For any graph G, the total degree of G is even.
Solution:
The total degree of G is even, for any graph G.
Steps:
For any graph G, the total degree is even
A
B
A, B turns into B, A
(b.) For any isosceles triangle T, the base angles of T are equal.
Solution:
The base angles of T are equal, for any isosceles triangle T.
Steps:
The base angles of T are equal, for any isosceles triangle T.
A
A, B turns into B, A
B
c. p is even, for some prime number p.
The answer is put as an existential-informal
statement, where if it were put in a formal
statement if would look like:
∃𝑝 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝.
 d. f is not differentiable, for at least one continuous
function f.
The answer is put as an existential-informal statement,
where if it were put in a formal statement it would look
like:
∃𝑓 𝑖𝑠 𝑛𝑜𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒, 𝑓𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢
𝑠
𝑓𝑢𝑛𝑐𝑡𝑖o𝑛 𝑓.
PREDICATES AND QUANTIFIED
STATEMENTS II
In each problem determine whether the
proposed negation is correct. If it is not,
write a correct negation.
The proposed negation is incorrect, because the
negation of “for all” is not “for all and also the
negation of an if-then statement is not an if-then
statement.
The correct negation should be:
Negation: There exists a real number x 1 and
x 2 , such that
x 1 2 ≠ x 2 2 and x 1 ≠ x 2 .
(∀x) x for all of u,if x is in Discrete Mathematics,then x is lower
case.
This statement is false because the lower case letter u is
usually used to denote elements of a set, therefore this means
that the upper case letter u can also be seen with in Discrete
Mathematics. In order to make this true the statement should
be:
Some occurrences of the letter u in Discrete
Mathematics are not lowercase.
Formal Notation: (∃x) for some of u,such that x is a lowercase
letter in Discrete Mathmatics
BIBLIOGRAPHY
 Epp, Susanna S. Discrete Mathematics With Applications 4 th
Edition. Boston, MA: Brooks/Cole Publishing Company, 2004.
Print.
 Brent, Joseph. Charles Sanders Pierce: A Life . Bloomington,
Indiana: Indiana University Press, 1998. Print.