Discrete Math – Logic Unit

1. Discrete Math – Logic Unit Jill Hubbard Tualatin High School

2. Oregon Department of Education approved discrete math advanced knowledge and skills • D.8 Logic: Understand the fundamentals of propositional logic, arguments, and methods of proof. • D.8.1 Use truth tables to determine truth values of compounded propositional statements. • D.8.3 Determine whether two propositions are logically equivalent. • D.8.5 Construct logical arguments using laws of detachment (modus ponens), syllogism, tautology, and contradiction

3. Materials Needed • Logisim free logic simulator • http://sourceforge.net/projects/circuit/ • Access to a computer

4. First some vocabulary we’ll see • Proposition • Compound Propositions • Primitive propositions • Logical Operators • Truth Table • Conjunction (AND) • Disjunction (OR) • Negation (NOT) • What other logical operators do you know? • Tautology • Contradiction

5. Propositions • A proposition is a statement that is either true or false • 2+2=4 (true) • New York City is in Oregon (false) • Today is Friday (its either true or false) • Do your homework (not a proposition) • It’s raining outside (probably true in Oregon!)

6. Compound and Primitive Propositions • Compound Propositions are propositions that are composed of sub-propositions connected together in various ways • roses are red and violets are blue • Mark is smart or he studies a lot • Primitive propositions can not be broken down into simpler propositions • If it’s not a composite proposition, it’s a primitive proposition

7. What do you think this means? • The truth value of a compound proposition id determined by the truth values of it sub-propositions together with the way in which they are connected • John is short OR John is tall • If either part is true, the entire proposition is true • John is short AND John is tall • Both parts have to be true for the entire proposition to be true.

8. p p q p * q q Basic Logical OperatorsConjunction • Conjunction: p q / (p * q) • Read as p AND q • A truth table is used to shows how a logical operator works • Use Logisim to model the AND operator and fill in the truth table • Based on your truth table, write a definition of how the AND operator works AND

9. Basic Logical OperatorsDisjunction • Disjunction: p q / (p + q) • Read as p OR q • A truth table is used to shows how a logical operator works • Use Logisim to model the OR operator and fill in the truth table • Based on your truth table, write a definition of how the OR operator works p p q p + q OR q

10. Basic Logical OperatorsNegation • Negation: ¬p / !p • Read as NOT p • Use Logisim to model the NOT operator and fill in the truth table • Based on your truth table, write a definition of how the NOT operator works • Let p be the statement 2+2 = 5 (true or false) • Give an example of a negation of this statement ¬ p !p p

11. NAND NOR XNOR XOR Other Logical Operations (NAND, NOR, XOR, XNOR) • Use Logisim to model these operators • Create a truth table for each operator • Based on your truth table, write a definition of how the operator works • XOR: read as exclusive OR • XNOR: read as exclusive NOR

12. Relationships between logical operators • People are related to each other right? So are logic gates. But you need to figure them out! • What the relationship between the AND gate and the NAND gate • What the relationship between the OR gate and the NOR gate • What the relationship between the OR gate and the XOR gate. What makes it so exclusive anyway? • What the relationship between the XOR gate and the XNOR gate

13. Truth Tables for Compound Propositional Statements • Create a truth table for the following compound propositional statement: (p q) (¬ p q) (p * q) + (!p * q) • Use Logisim to model this statement • Run the simulator to make it matches your truth table. If it doesn’t, you made a mistake and must figure out how to fix it! • From here on out, let’s use an engineer’s nomenclature mathematicians nomenclature engineers nomenclature

14. What the model looks like

15. The Mathematicians Method

16. The Engineers Method (p * q) + (!p * q) • Sum of products form • Each term produces a true expression • So, the expression is true when • p is true AND q is true OR • p is false (!p) AND q is true • Everywhere else, the expression is false • Engineers are lazy!

17. Tautologies • Tautologies are propositions that are always true no matter what the truth values if their variables are • (p + !p) is a tautology • Why? If p is true, then !p is false and visa versa • The OR logical operator states that if either p or !p is true, the result is true. One of those MUST be true • Use the simulator to build this and test it.

18. Contradictions • Contradictions are propositions that are always false no matter what the truth values if their variables are • (p * !p) is a tautology • Why? If p is true, then !p is false and visa versa • The AND logical operator states that if either p or !p is false, the result is false. One of those MUST be false • Use the simulator to build this and test it.

19. Simulate & Fill In the Truth Table

20. Tautologies & Contradictions • Let p stand for the statement “It is cold outside” • Then !p means it is not cold outside • The proposition, p * !p would mean that it is cold outside and it is not cold outside. Clearly, always false and therefore a contradiction • The proposition, p + !p would mean that it is cold outside or it is not cold outside. Clearly, always true and therefore a tautology

21. Tautologies & Contradictions • Is the following proposition a tautology or a contradiction? p + !(p * q) • Prove using a truth table • Prove using the Logisim simulator

22. Logical Equivalence • Two propositions are logically equivalent if they have identical truth tables • How is it possible that two propositions can have the same truth table? Take for example the following 2 propositions: !(p * q) !p + !q

23. Logical Equivalence !(p * q) !p + !q

24. Logical Equivalence • When engineers design circuits, the goal is to create designs that work robustly and are cheap • The fewer gates engineers use, the cheaper the design will be • Therefore, it is critical that engineers simplify their design to create cheap but equivalent logic • Engineers use logic synthesizers to help them do the task of logic simplification.

25. Engineering Applications • Engineers use 1’s and 0’s instead of True and False • 1 means True and 0 means False • All truth tables therefore use 1’s and 0’s • Computers use the binary number system • When engineers create a design, their first job is to determine the interface for their design (the inputs and outputs needed to get the job done).

26. Number SystemsDecimal – base 10 • Remember back a long time ago when you were learning how to count? • Our number system only has 10 symbols (0-9). So how do we represent the number after 9? • Each number had a place value (Each place value is a power of 10 (base 10) • You multiply each number with its place value and then added them all together. • Now you just take it for granted!

27. Decimals Numbers - Base 10

28. Number SystemsBinary – Base 2 • Computers use the binary number system or base 2. Base 2 uses only 2 numbers (1 and 0). This makes things very simple. But how do we represent numbers greater then 1? • Just like base 10(decimal), each number had a place value. Each place value is a power of 2 (base 2) • You multiply each number with its place value and then added them all together.

29. Binary Numbers - Base 2

30. Counting in Binary is EASY!

31. Applications of logic – 7 segment display project outA YOUR LOGIC outB in2 outC in1 outD outE in0 outF outG Note: A “1” on outA turn on the A segment of the 7 segment display, and a “0” on outA turns segment A off. The same is true for segments B through G

32. Applications of logic – 7 segment display project • Let’s use what we know to design logic that drives a 7-segment display. • All inputs and outputs are binary numbers (1’s and 0’s) • If your inputs are all 000, your 7 segment display should display the number 0 • If your inputs are all 111, your 7 segment display should display the number 7 • All input number encodings should work (0-7)

33. Let’s Make Truth Tables • Demo the truth table for outA • Let students figure out equations for outB-outG • Review your equations with your instructor

34. Lets’ Simplify our logic • Show students how to create K-Maps to create equivalent logic with less gates

35. Let’s use the logic simulator to make and test our design • Students must test their design to make sure they work • If they do not work, they must debug their design