Discrete Mathematics

1. Discrete Mathematics Carlos Aguilar Jose De Jesus Melendez Angie Rangel Drake Mackenzie Jain

2. Sections • Section 2.4: Application: Digital Logic Circuits # 34 • Section 3.1: Predicates and QuantifiedStatements I # 16, 21 • Section 3.2: Predicates and Quantified Statements II #14, 38

3. 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.

4. Terms/Symbols • BiconditionalStatement (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.

5. Section 2.4 Application: Digital Logic Circuits NOR Operator Notation: P ↓ Q (meaning neither P nor Q) ≡ ~ (P ˅ Q) 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.

6. 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.

7. Solutions 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

8. 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 ↓

9. 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

10. 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

11. Universal Quantifiers(Discrete mathematics section 3.1) By: Jose Melendez

12. Before That Predicates & Quantifiers & Charles S. Pierce

13. 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

14. 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

15. 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 (Epp97)     Universal Existential (For all) (There Exists)

16. 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

17. 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

18. 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

19. 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

20. More 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

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

22. 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.

23. Existential Quantifier and Formal Statement : ( • A Existential statement includes the quantifier ( ) = “some” Formal Statement: b a gamer, such that b is boy.

24. 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.

25. 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.

26. Predicates and Quantified Statements I

29. 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:

30. 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 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑓𝑢𝑛𝑐𝑡𝑖o𝑛 𝑓.

31. Predicates and Quantified Statements II In each problem determine whether the proposed negation is correct. If it is not, write a correct negation.

32. 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 ≠ x22 and x1 ≠ x2 .

33. (∀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

34. bibliography • Epp, Susanna S. Discrete Mathematics With Applications 4th Edition. Boston, MA: Brooks/Cole Publishing Company, 2004. Print. • Brent, Joseph. Charles Sanders Pierce: A Life. Bloomington, Indiana: Indiana University Press, 1998. Print.