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 Access to a computer
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
Have students guess what they think these works mean.
The goal is to get them thinking about the words apart from math and then add the math definitions on top of that

5. Propositions A proposition is a statement that is either true or false
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.
Open discussion. Try to get to the fact that in order to determine if the compound prop is true, we’ll need to break it up into its pieces and then figure out how we’re connecting the pieces

8. Basic Logical Operators Conjunction
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
p * q
AND p q
p q F T
Share out the definitions and give student an official definition
If p and q are true, then p AND q is true, otherwise its false

9. Basic Logical Operators Disjunction
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 p q
p q F T
Share out the definitions and give student an official definition
If either p or q are true, then p OR q is true, otherwise its false OR
If p AND q are false, the p OR q is false. Otherwise, its true

10. Basic Logical Operators Negation
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 p
¬ p F T
Share out the definitions and give student an official definition
If p is true then not p is false
If p is false then not p is true
The statement 2+2 = 5 is false. Therefore, its negation must be true
2+2 is not equal to 5 (that’s true) or It is not true that 2+2 = 5 (also true)

11. 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

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
NAND is not AND (hence the bubble on the output of the gate.) The truth table outputs are flipped (notted)
NOR is not OR (hence the bubble on the output of the gate.) The truth table outputs are flipped (notted)
An XOR looks like an OR operator except the case when both inputs are true results in an false output (its excluded)
XNOR is not XOR (hence the bubble on the output of the gate.) The truth table outputs are flipped (notted)

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
File name of source: combo logic

15. The Mathematicians Methodp q !p
p * q
!p * q
(p * q) + (!p * q) F T

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 Tablep
p * !p F T p
p + !p F T

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 F T F T !(p * q) !p + !q p q p * q !(p * q) p q !p

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 Systems Decimal – 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
10,000’s Place
1000’s Place
100’s
Place
10’s
1’s Place
Number 3
3*1 = 3 2
(2*10+3*1)= 23 1
(1*100)+(2*10)+(3*1)= 123

28. Number Systems Binary – 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 1 (1*4)+ (1*2) = 6 (1*16) + (1*4)+ (1*2)
16’s Place
8’s Place
4’s
Place
2’s
1’s Place
Number 1
(1*4)+ (1*2) = 6
(1*16) + (1*4)+ (1*2)
Or simply
= 22
= 31

30. Counting in Binary is EASY!
Binary Number
Decimal Equivalent
zero 1
one
two
three
four
five
six
seven

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