Carlos Aguilar

Jose De Jesus Melendez

Angie Rangel

Drake Mackenzie Jain

DISCRETE MATHEMATICS

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

SECTIONS

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.

TERMS/SYMBOLS

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)

SECTION 2.4 APPLICATION: DIGITAL LOGIC CIRCUITS

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.

SHOW THAT THE FOLLOWING LOGICAL EQUIVALENCES HOLD FOR THE PEIRCE

ARROW ↓, WHERE P↓Q ≡~(P˅Q).

a. ~P ≡ P ↓ P~P ≡ ~ (P ˅ P) by the idempotent law for ˅≡ P ↓ P by definition of ↓

b. P ˅ Q ≡ (P ↓ Q) ↓ (P ↓ Q)≡ ~(P ↓ Q)≡ ~(~P ˄ ~Q) by definition of ↓≡ P ˅ Q by the double negation law

SOLUTIONS

c. P ˄ Q ≡ (P ↓ P) ↓ (Q ↓ Q)P ˄ Q ≡ ~(~(P ˄ Q)) by the double negation law≡ ~(~P ˅ ~Q) by De Morgan’s law≡ ~((P ↓ P) ˅ (Q ↓ Q))≡ (P ↓ P) ↓ (Q ↓ Q) by definition of ↓

d. Write P → Q using Peirce arrows only.

P → Q ≡ (~P ↓ Q) ↓ (~P ↓ Q)≡ ~ (~P ↓ Q)≡ ~ (~ (~P ˅ Q)) by definition of ↓≡ ~ (P ˄ ~Q) by the double negation law≡ ~P ˅ Q by De Morgan’s Law≡ P → Q

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

By: Jose Melendez

UNIVERSAL QUANTIFIERS

(DISCRETE MATHEMATICS SECTION 3.1)

Predicates&

Quantifiers&

Charles S. Pierce

BEFORE THAT

Charles Sanders Pierce1839 – 1914Sometimes known as “Father of

Pragmatism”B.A., B.Sc., M.A. from Harvard

UniversityMathematician, 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 fi lling up of its details.” – Charles Pierce

THE MAN BEHIND THE LOGIC

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

PREDICATES

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 Existential (For all) (There

Exists)

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

FOCUS: UNIVERSAL QUANTIFIERS

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? - is not equal to 3

3. Statement? - real number x, is not equal to 3

HOW IT IS DONE

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

EXAMPLES

5. The number 2,147,581,953 is not equal to the square

of any integer.

integer x, 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, is not equal to -1

MORE EXAMPLES

A Universal statement includes the quantifier ( ∀ ) = “all”For Example:∀x, if x is chocolate, then x tastes good

UNIVERSAL STATEMENT: ( ∀ )

x, if x is tastes good, then x is chocolate.

Negation is….

x, such that x tastes bad then x is not chocolate.

UNIVERSAL EXAMPLE

EXISTENTIAL QUANTIFIER AND FORMAL STATEMENT : (

A Existential statement includes the quantifier

( ) = “some”

Formal Statement: b a gamer, such that b is boy.

b a boy, such that b is gamer.

Negation is…

b is a boy, if b is a gamer then b is a boy.

EXISTENTIAL EXAMPLE

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 .

INFORMAL LANGUAGE OF LOGIC:

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 B

A, B turns into B, A

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:

 

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 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑓𝑢𝑛𝑐𝑡𝑖o .𝑛 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 x1 and x2, such that

x12 ≠ x2

2 and x1 ≠ x2 .

(∀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

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.

BIBLIOGRAPHY