CT 100 Week 4

1. CT 100 Week 4 Logic

2. Logic Continued • Quiz 4 Vocabulary • Quiz 4 Problems • Implication • Translation of English Statements to Symbolic Logic • Well-formed Boolean Expressions • Digital Logic

3. Quiz 4 Vocabulary • Equivalence • Contradiction • Conclusion • Law of excluded middle • Law of non-contradiction • Boolean Logic • Premise • Proposition • Syllogism • Symbolic logic • Tautology • Truth table • Definitions for the new terms are at the end of chapter 3

4. Quiz 4 Quiz Problems • Convert binary to base 10 • Convert base 10 to binary • Convert a sequence of characters to a sequence ASCII codes (numbers) • Convert a sequence of numbers representing characters in ASCII to a sequence of characters • Create a truth table for a Boolean expression • Show that 2 Boolean expressions are equivalent • Translate an English language statement into symbolic logic • Determine if a expression is a well formed Boolean expression

5. IMPLIES Truth Table

6. Implication • Conditionals • If P Then Q • P is the antecedent • Q is the consequence • Stoics • Philo of Megara • A conditional is false when and only when the antecedent is true and the consequence is false

7. Implication • Truth-functionality • The truth value of a compound statement is a (total) function based only on the truth values of its parts (which must be propositions) • Frege • Russell and Whitehead • P IMPLIES Q is equivalent to (NOT P) OR Q

8. Implication • References • www.maa.org/sites/default/files/images/upload_library/46/Pengelley_projects/truth.pdf • http://plato.stanford.edu/entries/conditionals/ • http://plato.stanford.edu/entries/dialectical-school/

9. English to Symbolic Logic Translation • Let A represent the simple statement “Alex is a computer science major” • Let B represent the simple statement “Alex takes CS 120” • Let C represent the simple statement “Alex is a Biology major” • Let D represent the simple statement “Alex takes BIO 105”

10. Translate the Following Statements into Symbolic Logic Expressions and Build the Truth Tables for the Expressions • Alex is a computer science major and Alex is a biology major • Alex is a computer science major or Alex is a biology major • Alex is not a computer science major • Alex does not take BIO 105 • If Alex is a computer science major then Alex takes cs 120 • If Alex takes BIO 105 then Alex is a biology major

11. Translate the Following Statements into Symbolic Logic Expressions and Build the Truth Tables for the Expressions • If Alex takes CS 120 then Alex is a computer science major • If Alex is a computer science major or Alex is a biology major then Alex takes BIO 105 • If Alex does not take BIO 105 then Alex is not a biology major

12. Well-formed Expressions • Syntactically correct boolean expressions • Rule 1 • Each single letter is a well-formed expression • True is a well-formed expression • False is a well-formed expression

13. Well-formed Expressions • Assume P and Q are well-formed expressions • Rule 2 • P and Q is a well-formed expression • P or Q is a well-formed expression • P IMPLIES Q is a well-formed expression • P ≡ Q is a well-formed expression • NOT P is a well-formed expression • Rule 3 • ( P ) is a well-formed expression

14. Well-formed Expressions • Are the following expressions well-formed expressions? • P and (NOT Q) • (A OR B) AND (C OR D) • (R AND S) (NOT W) • (NOT (A AND B)) IMPLIES (B OR C) • (NOT P) IMPLIES (NOT Q) • NOT IMPLIES B OR C

15. Digital Logic • Building Blocks Digital Computer Hardware • Logic Gates • And gate • Or gate • Not gate • Must be implemented with physical devices • For example transitors • The are a low level abstraction

16. And Gate

17. Or Gate

18. Not Gate

19. Adding three bits 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 +0 +1 +0 +1 +0 +1 +0+1 00 01 01 10 01 10 10 11

20. Truth Table for One Bit Adder0 means False and 1 means True