1. IB Computer Science Computer Organization
Part 4 – Digital Logic, Logic Gates, and Boolean Expressions

2. What the IB curriculum says…

3. Boolean Algebra

4. Boolean Algebra A variant of algebra where values can only be
TRUE (1) or FALSE (0)
Core operators are not addition, multiplication, etc.
Instead we use
AND, OR, and NOT. Others covered are NAND, NOR, and XOR.
Others exist, but aren’t covered on the IB test.

5. Boolean Notation Variables are what you’re used to: X, Y, A, B, etc.
Values can only be 0 or 1 (or True/False)
Core functions notation:
NOT = A’ (A prime) – sometimes shown as 𝐴
AND looks like multiplication: AB, or A·B, or sometimes A˄B
OR looks like addition: A+B, or sometimes A˅B
Let’s look at the details for each one…

6. NOT operator NOT just means the opposite Notation reminder: A’
Since the only values are 0 and 1:
0’ = 1
1’ = 0
So if A = 0, A’ = 1, and if B = 1, B’ = 0
A TRUTH TABLE for NOT is here:
In circuit design, we use a NOT gate, also called and inverter (the circle means flip or invert):
So this means we can trace through a circuit… A A’ 1

7. AND operator AB AND usually has multiple inputs
Notation reminder: AB, or A·B, or A˄B
Output is TRUE (1) only if both inputs are TRUE (1)
AB = 1 only if A=1 and B=1
The TRUTH TABLE is:
The circuit symbol is: A B A B AB 1 AB

8. OR operator A+B OR usually has multiple inputs
Notation reminder: A + B, or sometimes A˅B
Output is TRUE (1) if either or both inputs are TRUE (1)
AB = 1 if A=1 OR B=1 (or both =1)
The TRUTH TABLE is:
The circuit symbol is: A B A B
A+B 1
A+B

9. IB Test Specifics
On the IB Test, you don’t have to remember the symbols for these, but you should learn them anyways for college
Just remember the types of logic
AND OR
NOT
Coming later…
NAND
NOR
XOR
And draw them as:
AND

10. Boolean Expressions

11. Boolean Expressions
…like algebraic expressions in some ways. We use them to define the result of processing inputs with logic gates
Parenthesis can be used to define precedence
The priority for precedence is:
Work expressions inside parenthesis first
NOT
AND OR

12. Boolean expression examples
Example 1: If I did my homework (A) AND didn't play video games(B represents playing video games), then I get ice cream for dessert. AB' represents whether I get ice cream
AB’ take the opposite of B first, then AND with A
Example 2: If I did my homework (B) AND fed the dog (C), OR I babysat my sister (A), then I can watch Daredevil on Netflix. A + BC represents whether I can watch Netflix
A+BC take AND of BC first, then OR with A

13. More practice, the other way
Write on paper, the equations of:
Y=??
X=??
Draw a diagram using all three types of symbols, and have your partner solve it by writing the equation

14. Example 1: Solving an equation for a specific set of inputs
For the equation: (A+B)’ + CD(E+FG’)
Solve where A=1, B=0, C=1, D=1, E=0, F=1, and G=0
What does the above expression result in?
Note you can show the result in:
Words
A TRUTH TABLE
A Boolean Expression as above
A Circuit diagram
These are all equivalent solutions, just different ways to look at it
Let’s create a TRUTH TABLE, and a Circuit for the above
We’ll work this one together on the Whiteboard… TURN OFF YOUR MONITORS, DON’T LOOK AHEAD IN THE SLIDES

15. Whiteboard:(A+B)’ + CD(E+FG’) Solve where A=1, B=0, C=1, D=1, E=0, F=1, and G=0
Truth Table (add columns for the expression parts)
Circuit diagram (work from inside out, and down) A B C D E F G G'
FG'
etc 1

16. Example 1 solution for review
(A+B)’ + CD(E+FG’)
A TRUTH TABLE if A=1, B=0, C=1, D=1, E=0, F=1, and G=0
Circuit Diagram A B C D E F G
A+B
(A+B)’ G’
FG’
(E+FG’) CD
CD(E+FG’)
(A+B)’ + CD(E+FG’) 1
You can fill out all of the other combinations too, but that’s a lot of work!

17. Example 2 A car has 3 sensors Create:
K is TRUE if the key is in the ignition
D is TRUE if the door is open
L is TRUE if the lights are on
Output Buzzer B should be true if:
The lights are on when the key is not in the ignition, or when the door is open and the key is in the ignition
Create:
A Boolean expression
A FULL truth table showing all 8 combinations of inputs
(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0), etc.
A circuit diagram using logic gates
TURN OFF YOUR MONITORS AGAIN. DON’T LOOK AT THE NEXT SLIDE. IT’S THERE FOR REVIEW WHEN YOU'RE RESTUDYING THIS
Work with a partner or two

18. Example 2 solution for review
LK’ + DK = B L K D B 1

19. Practice tools for logic gates
In the Unit 2 folder on the Google drive, open the Logic Circuit folder and install Cedar Logic Simulator (CedarLSSetup.exe)
Use it to drag and drop gates, connect them, and test out how they work. Focus on the top 3 categories of gates:
Basic Gates (AND, OR, and later NAND, NOR, XOR)
Invert and Connect (INVERTER = NOT, From and To with the Oscilloscope feature)
Input and Output (On/Off Switches, Square Wave, LED for output)
Use View, OScope to use the Oscilloscope feature with square waves
Requires Using the To block to measure something on the scope
Double click the To block to relabel it
Double click the Square Wave block to change its frequency

20. Cedar Logic Sim (AND gate)
We’ll build this together
so you can become familiar
with the tool

21. Exercise: Build Ex 2 in Cedar
3 Inputs: Lights, Key, Door (Connect to on/off switches)
One Output: B for Buzzer (Connect to an LED)
Add in and connect up the gates
Test it out by flipping the switches to see if the LED comes on when appropriate (simulates the buzzer going off)

22. Example 3 Convert this circuit diagram to a Boolean expression
Create and analyze this in Cedar
Create a full TRUTH TABLE showing X for all 8 input combos
Also, if A=1, B=1 and C = 0, what is X?
TURN OFF YOUR MONITORS AGAIN. DON’T LOOK AT THE NEXT SLIDE. IT’S THERE FOR LATER REVIEW

23. Example 3 solution for review
If A=1, B=1 and C = 0, what is X?
Working through:
AB=1
B’C = 0
(B’C)’=1
AB+(B’C)’ = 1
(AB+(B’C)’)’ = 0 A B C X 1

24. Simplifying Expressions
If Q = (A’ + B)’C + AC’
Then we have this TRUTH TABLE
Looking at the table, we see that Q is 1 when A is 1, except if B AND C are both 1
Thus, we can simplify to:
Q=A(BC)’
What do the circuits look like?
Which is simpler?
Which would cost less to implement using gates? A B C Q 1

25. Other ways to simplify
The following are the basic Boolean theorems. They are used to simplify boolean expressions. You'll cover these in detail in college.
Idempotent Law
A * A = A
A + A = A
Associative Law
(A * B) * C = A * (B * C)
(A + B) + C = A + (B + C)
Commutative Law
A * B = B * A
A + B = B + A
Distributive Law
A * (B + C) = A * B + A * C
A + (B * C) = (A + B) * (A + C)
Identity Law
A * 0 = 0     A * 1 = A
A + 1 = 1     A + 0 = A
Complement Law
A * A' = 0
A + A' = 1
Involution Law
(A')' = A
DeMorgan's Law
(A * B)' = A' + B'
(A + B)' = A' * B'

26. One Simplification Example
Simplify: C + (BC)':
Expression Rule(s) Used
C + (BC)' Original Expression
C + (B' + C') DeMorgan's Law (BC)' = B' + C'.
(C + C') + B' Commutative, Associative Laws.
True + B' Complement Law.
True Identity Law.
Lots of examples online to see more:
search for Boolean Expression Simplification

27. NAND and NOR gates

28. NAND and NOR are special
NAND is just the inversion of AND
NOR is just the inversion or OR
The electronics (silicon chip design) is typically faster than AND and OR, and cheaper to build (less transistors)
Interestingly, any circuit can be built using just NAND and NOR gates

29. NAND gates: X AND Y, InvertedXY
(XY)’ 1
NAND = (XY)’
Symbol and TT:
Where the dot means take the opposite of AND
If the inputs are X and Y, then the output is (XY)’
Looking at the TT: the output is the same as: X’ + Y’ X Y X’ Y’
X’+Y’
(XY)’ 1

30. NOR gates: X or Y, inverted
X+Y
(X+Y)’ 1
NOR = (X+Y)’
Symbol and TT:
Where the dot means take the opposite of OR
If the inputs are X and Y, then the output is (X+Y)’
Looking at the TT: the output is the same as: X’Y’ X Y X’ Y’
X’Y’
(X+Y)’ 1

31. Alternative/equivalent gates
As we saw above from the Truth Tables:
NAND = (XY)’ = X’ + Y’
NOR = (X + Y)’ = X’Y’
AND = XY = (X’ + Y’)’
OR = X + Y = (X’Y’)’

32. AND/OR, NAND/NOR equivalency
Many expressions are written like:
AB+CD - a Sum of Products, or SOP
(A+B)(C+D) - a Product of Sums, or POS
Which yielded circuits using AND and OR gates
We’ve said these gates are more expensive and slower than NAND and NOR gates

33. AND/OR, NAND/NOR equivalency
But we can also use NAND and NOR gates
For a Sum of Products (SOP), we can just use NAND gates
Replace with
Because:
NAND = (XY)’ = X’ + Y’
The negatives (NOTs) cancel,
Leaving AND / OR combo: =

34. AND/OR, NAND/NOR equivalency
Similarly for Product of Sums, or POS, we can just use NOR gates
Because:
NOR= (X + Y)’ = X’Y’
Replace with
The negatives (NOTs) cancel,
Leaving OR / AND combo: =

35. AND/OR, NAND/NOR equivalency
Summary
If you have an SOP expression F = AB+A’C+B’D you can use AND / OR gates, or JUST NAND gates
If you have a POS expression F = (A+B’)(C’+D) you can use OR / AND gates, or JUST NOR gates
Also use inverters (NOTs) when needed

36. NAND Examples for SOP

37. NOR examples for POS

38. Exclusive OR (XOR)

39. XOR operator XOR is TRUE if either input is TRUE, but NOT BOTH INPUTS
Notation: A B
Output is TRUE (1) only if one input is TRUE (1)
A B = 1 only if A=1 OR B=1
The TRUTH TABLE is:
The circuit symbol is: A B
XOR 1

40. XOR Example Create a truth table for A XOR (B OR C) Solution:
Fill out the ABC values (000, 001, 010, 011, 100, 101, etc.)
Do B OR C first – the stuff in parenthesis
Now do column A XOR'd with column (B OR C)
Answer is True(1) only if one or the other column is 1 (but not both) A B C
(B OR C)
A XOR (B OR C) 1

41. Summary of Gates Venn diagrams overlap with Math, Logic, Reason, TOK
𝐴 is same as A’ i.e., NOT

42. Boolean Algebra Theorems (for reference, not on tests)
Idempotent Law
A * A = A
A + A = A
Associative Law
(A * B) * C = A * (B * C)
(A + B) + C = A + (B + C)
Commutative Law
A * B = B * A
A + B = B + A
Distributive Law
A * (B + C) = A * B + A * C
A + (B * C) = (A + B) * (A + C)
Identity Law
A * 0 = 0     A * 1 = A
A + 1 = 1     A + 0 = A
Complement Law
A * ~A = 0
A + ~A = 1
Involution Law
~(~A) = A
DeMorgan's Law
~(A * B) = ~A + ~B
~(A + B) = ~A * ~B
You can use these to simplify expressions
manually.
In college, you’ll do this, along with something called Karnaugh maps (K Maps)