Discrete Math – Logic Unit

Jill Hubbard

Tualatin High School

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

Materials Needed

• Logisim free logic simulator– http://sourceforge.net/projects/circuit/

• Access to a computer

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

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!)

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

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.

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

p q p q

F F

F T

T F

T T

p

qp qp * q

AND

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 q p q

F F

F T

T F

T T

p

qp qp + q

OR

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

F

T

p ¬ p !p

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

NAND NOR XNOR XOR

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

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

engineers nomenclature

mathematicians nomenclature

What the model looks like

The Mathematicians Method

p q !p p * q !p * q (p * q) + (!p * q)

F F T F F F

F T T F T T

T F F F F F

T T F T F T

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!

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.

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.

Simulate & Fill In the Truth Table

p p * !p

F

T

p p + !p

F

T

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

Tautologies & Contradictions

• Is the following proposition a tautology or a contradiction?p + !(p * q)

• Prove using a truth table

• Prove using the Logisim simulator

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

Logical Equivalence

p q p * q !(p * q)

F F F T

F T F T

T F F T

T T T F

p q !p !q !p + !q

F F T T T

F T T F T

T F F T T

T T F F F

!(p * q) !p + !q

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.

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).

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!

Decimals Numbers - Base 10

10,000’s Place

1000’s Place

100’s

Place

10’s

Place

1’s Place

Number

0 0 0 0 3 3*1 = 3

0 0 0 2 3 (2*10+3*1)= 23

0 0 1 2 3 (1*100)+(2*10)+(3*1)= 123

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.

Binary Numbers - Base 2

16’s Place

8’s Place

4’s

Place

2’s

Place

1’s Place

Number

0 0 1 1 0 (1*4)+ (1*2) = 6

1 0 1 1 0 (1*16) + (1*4)+ (1*2)

Or simply

16+4+2 = 22

1 1 1 1 1 16+8+4+2+1 = 31

Counting in Binary is EASY!

Binary Number Decimal Equivalent

0 0 0 zero

0 0 1 one

0 1 0 two

0 1 1 three

1 0 0 four

1 0 1 five

1 1 0 six

1 1 1 seven

Applications of logic – 7 segment display project

YOUR LOGIC

in2

in1in0

outAoutBoutCoutDoutEoutFoutG

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

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)

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

Lets’ Simplify our logic

• Show students how to create K-Maps to create equivalent logic with less gates

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