Project Part 1

1. Project Part 1 Aaeron Jhonson-Whyte • Akuang Saechao • Allen Saeturn

2. Section 2.1, Problem 29 • Use De Morgan’s laws to write negations for this statement: • This computer program has a logical error in the first ten lines or it is being run with an incomplete data set.

3. Section 2.1, Problem 29: Solution • To first solve this problem, we must understand the definition of a statement and statement form. • Definition of a statement : • A statement (or proposition) is a sentence that is true or false but not both. • Definition of statement form : • A statement form (or propositional form) is an expression made up of statement variables (such as p,q, and r) and logical connectives (such as ∼,∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables. • Definition of disjunction : • If p and q are statement variables, the disjunction of p and q is “p or q,” denoted p ∨ q. It is true when either p is true, or q is true, or both p and q are true; it is false only when both p and q are false.

4. Section 2.1, Problem 29: Solution • We know that the statement is in statement form because it is two variables separated by the logical connective “or” and we can define the variables being P and Q. • This computer program has a logical error in the first ten lines, OR it is being run with an incomplete data set. • P = has a logical error in the first ten lines • ∨ =OR • Q = is being run with an incomplete data set • Which is written as P ∨Q.

5. Section 2.1, Problem 29: Solution • The problem asks for this statement to be negated through De Morgan’s Law, and that can be done by understanding negation and De Morgan’s Law. • Definition of negation: • If p is a statement variable, the negation of p is “not p” or “It is not the case that p” and is denoted ∼p. It has opposite truth value from p: if p is true, ∼p is false; • If p is false, ∼p is true. • Definition of De Morgan’s Law: • The negation of an and statement is logically equivalent to the or statement in which each component is negated. • The negation of an or statement is logically equivalent to the and statement in which each component is negated.

6. Section 2.1, Problem 29: Solution • So, P ∨ Q then becomes ~P ∧ ∼Q. • ~P= does not have a logical error in the first ten lines • ∧ =AND • ~Q = is not being run with an incomplete data set • The solution: • This computer program does not have a logical error in the first ten lines, ANDit is not being run with an incomplete data set.

7. Section 2.4, Problem 32 • The Boolean expression for the circuit in Example 2.4.5 is shown below. Find a circuit with at most three logic gates that is equivalent to this circuit. • (P ∧ Q ∧ R) ∨ (P ∧ ∼Q ∧ R) ∨ (P ∧ ∼Q ∧ ∼R)

8. Section 2.4, Problem 32: Solution • Definition of Distribution: • p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r ) • p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r ) • Definition of Negation: • If p is a statement variable, the negation of p is “not p” or “It is not the case that p” and is denoted ∼p. It has opposite truth value from p: if p is true, ∼p is false; • If p is false, ∼p is true. • Definition of Identity: • p ∧ t ≡ p             • p ∨ c ≡ p

9. Section 2.4, Problem 32: Solution • The circuit from the resultant Boolean expression is:

10. Section 2.4, Problem 33a • Show that for the Sheffer stroke |: • P ∧ Q ≡ (P | Q) | (P | Q)

11. Section 2.4, Problem 33a: Solution • Definition of Sheffer stroke |: • p | q ≡ ∼ (p ∧ q) • Definition of De Morgan’s Law: • The negation of an and statement is logically equivalent to the or statement in which each component is negated. • The negation of an or statement is logically equivalent to the • Definition of Idempotent: • p ∧ p ≡ p                • p ∨ p ≡ p

12. Section 2.4, Problem 33a: Solution • Therefore; • P ∧ Q ≡ (P | Q) | (P | Q)

13. Section 2.4, Problem 33b • Use the results of Example 2.4.7 and part (a) above to write P ∧ (∼Q ∨ R) using only Sheffer strokes.

14. Section 2.4, Problem 33b: Solution • Definition of Sheffer stroke |: • p | q ≡ ∼ (p ∧ q) • Definition of De Morgan’s Law: • The negation of an and statement is logically equivalent to the or statement in which each component is negated. • The negation of an or statement is logically equivalent to the • Definition of Idempotent: • p ∧ p ≡ p                • p ∨ p ≡ p

15. Section 3.3, problem 43 • The definition for limx→a f (x) = L: • For all real numbers ε > 0, there exists a real number δ > 0 such that for all real numbers x, • If a − δ < x < a + δ and x ≠ a, • then
L − ε < f(x) < L + ε. • Write what it means for limx→af (x) ≠ L. In other words, write the negation of the definition.

16. Section 3.3, problem 43 • Definition of ∀: • The symbol ∀ denotes “for all” and is called the universal quantifier. • Definition of ∃: • The symbol ∃ denotes “there exists” and is called the existential quantifier. • Definition of Negation of ∀: • The negation of a universal statement (“all are”) is logically equivalent to an existential statement (“some are not” or “there is at least one that is not”). • Definition of Negation of ∃: • The negation of an existential statement (“some are”) is logically equivalent to a universal statement (“none are” or “all are not”). • Definition of Limit: • The definition of limit of a sequence uses both quantifiers ∀ and ∃ and also if-then. • Solution: • For all real numbers ε ≤ 0, there exists a real number δ ≤ 0 such that for all real numbers x, • if a + δ ≤ x ≤ a − δ and x = a • then L + ε  ≤ f(x) ≤ L − ε.

17. Augustus De Morgan • Augustus De Morgan was born in Madurai, Madras Presidency, India in 1806. Augustus was a mathematician and professor. Augustus wrote mathematical texts such as Elements of Arithmetic (1830), Penny Cyclopedia (1838) where he created and coined ‘mathematical induction’, Trigonometry and Double Algebra (1849), a geometric interpretation of complex numbers, and Formal Logic (1847). • De Morgan couldn’t receive his graduate education and fellowship because he would not sign required theological forms that were required by Trinity College, De Morgan’s school at the time. De Morgan founded many symbolic logic as it with George Boole. De Morgan is best known as the creator of De Morgan’s Laws: • NOT (A AND B) = (NOT A) OR (NOT B) • NOT (A OR B) = (NOT A) AND (NOT B) • Also Seen As • ~ (A & B) = (~A) V  (~B) • ~(A V B) = (~A) & (~B) • When De Morgan was 62 years of age, his son George, died. Shortly after the death of his son, the death of a daughter followed. Fives years after his resignation from University College, De Morgan died of nervous prostration on March 18, 1871.

18. Bibliography • Epp, Susanna S.. Discrete mathematics with applications. 4th ed. Boston, MA: Brooks/Cole, 2011. Print. • "Cambridge SEARCHES." De Morgan's INFO. N.p., n.d. Web. 11 Mar. 2014. <http://venn.lib.cam.ac.uk/cgi-bin/search.pl?sur=&suro=c&fir=&firo=c&cit=&cito=c&c=all&tex=%22D823A%22&sye=&eye=&col=all&maxcount=50>. • "Who was Augustus De Morgan?." Who was Augustus De Morgan?. N.p., n.d. Web. 11 Mar. 2014. <http://www.demorgan.com/demorgan.htm>.