Going From Beginner To Expert On Computing Boolean Expressions and Finding Their Respective Logic Gates

1. Going From Beginner To Expert
On Computing Boolean
Expressions and Finding Their
Respective Logic Gates
Joshua Fernandes

2. The Outline Of This Tutorial
1. Basics Of Getting Started
2. Boolean Operators & Intro To Logic Gates
3. Truth Tables
4. Boolean Laws
5. Solving Boolean Algebra Problems
6. Making Logic Gates Using Boolean Algebra
7. Problems For You To Try

3. Basics Of Getting
Started
➔ Boolean Data Type
Boolean is a true/false value with
only 2 values being accepted for
this data type.
➔ 0’s & 1’s
In the Boolean algebra
application, the values being
used are 0’s and 1’s representing
binary values. Binary is the
language that the computer
speaks in and it what we speak
with to the computer.

4. Basics Of Getting
Started Pt.2
➔ Letters Used In Boolean
Algebra
The letters other that O
represents the different inputs
going into logic gates or truth
tables as a rule since the O
represent the output of the
Boolean operators
O
A
B
C
D

5. Boolean Operators & Intro To
Logic Gates

6. Logic Gates
- A Boolean operator is simply
an operation such as or & xor
that takes in either one or two
Boolean values and then
outputs one Boolean output
from that operation.
- There are 7 Boolean
operations that are used.
Boolean Operators
- A Logic gate is used to visually
represent a Boolean operation
with it representing both the
inputs going into the operation
and the output from the
operation.
- This is applied and used to
show what happens within a
computer system.

7. OR GATE
- The or operator is where for
two operators go in, their either
value is 1, the output is 1 else
0.
- For selecting fruit, the output is
still one as long as you pick
any amount of either bananas,
apples or both. The operator
doesn’t care what fruit is
chosen but it wants a fruit.

8. AND GATE
- The and operator is where for
two operators go in, if both
values are 1 then the output is
one else 0.
- For selecting fruit, the output is
one as long as you pick only
both a banana and an apple.

9. NOT GATE
- The not operator flips a value
so if the Boolean value is 0,
then it’s flipped to 1.
- For selecting fruit, the output is
anything but a banana or no
bananas. It’s like life or death,
their is no in-between.

10. XOR GATE
- The xor operator (exclusive or_
is where for when one input is
1 and the other is 0, the output
is 1 else 0.
- For selecting fruit, the output is
one as long as you pick only
both a banana and an apple. As
a weird analogy, think of this
gate as a homophobic gate
where the output is 1 only
when the inputs are the

11. NAND, NOR & XNOR
GATES
NAND
NOR
AND
XNOR
OR
XOR
THE NOT GATE
MIRROR
- The nand, nor and xnor gates are the direct
opposite of the and, or and xor gates with
the outputs of the nand, nor and xnor gates
being the opposite of their counterparts.
The XNOR Gate
The NOR Gate

12. NAND GATE
- The NAND is the foundation for
all the other logic gates seen
which is why it’s referred to as
the universal gate NAND
NOT
AND
OR
NAND

13. TRUTH TABLES
- A truth table is a visual representation of
all possible combination of inputs for
either one or multiple Boolean operations
and it shows the resulting output from
each operation. This can be used to even
find the operation combinated for the
result even if we don’t know exactly
which operations occurred.
- There are also 2 main types of truth
tables, one shows the operation in
isolation and the other encompasses
multiple operations.
It reveals the TRUE output
possibilities for each operation

14. TRUTH TABLES PT.2
The single operation table
The multiple operation table
- The multiple operation truth table is a more useful form of truth table since
it encompasses more information useful for both computer systems and
mathematical applications. Through the process of Boolean algebra, the
output is “(A ∧ (Ā + AB))” but it’s usually more complicated to find this.

15. Boolean Laws

16. Identity Laws
A ∧ 1 = A A + 0 = 0
Domination Laws
A + 1 = 1 A ∧ 0 = 0
These 2 laws compares 1 input to the Boolean values of 0 and 1.
- Tip 1: Double check to see if the operation you’re doing is an and or an or
since if you compute the wrong result, then the whole operation have the
wrong output.
- Tip 2: Make sure that you have the correct result because if you don’t,
then you may have done the inverse of this law which will affect the rest
of the calculation down the road.
These 2 laws are inverses of each other and their results are inverse so
please, DOUBLE CHECK YOUR WORK.

17. Negation Laws
Negation Law: This law gets an input and it’s inverse
to either apply the AND or OR
operation to find the result of this.
- Tip 1: This is cancelling out the
not A and A so when you are
done with these, you only take
the output if you need to do
further algebra.
Double
Negation Law:
This law cancels out 2 negative values
in the input if there are 2 or more not
operations in the input.
- Tip 1: If the amount of lines or
not values attached to the input
is odd, just eliminate all of them
and if it’s odd, then eliminate all
of them too but just leave one
still there to then apply other
laws within the Boolean
expression.
A practical way of thinking about
the Double Negation Law
Eliminates an input and it’s inverse
for a Boolean value.

18. Idempotent Laws
➔ This law deals with when
there are two of the same
input in the same
expression simplifying the
result to its original input.
➔ Tip 1: The great thing
about this law is even if
you mess up the type of
operation you execute, the
answer will be the same.
➔ A ∧ A = A
➔ A + A = A
It gets rid of multiples
of the same input
using the idempotent
law

19. Absorption Law
A ∧ (A + B) = A
A + (A ∧ B) = A
This law takes an expression
with 2 of the same input and
one of another input with there
being 2 DIFFERENT
operations separate each input
with the result being just the
input repeated twice. (This law
simplifies the expression
Tip 1: When checking to see if
this operation can be done to
an expression, if the operation
separating the 3 inputs are the
same, DO NOT APPLY THE
ABSORPTION LAW.

20. COMMUNICATIV
E LAWS
This law deals with when there is an
input of one time and input of the
other type and this law simply flips
the order of the inputs.
- Tip 1: This only reverses the
order of the inputs which can
be at time important so DON’T
SWITCH THE OPERATIONS.
A ∧ B = B ∧ A
A + B = B + A

21. Distributive Laws
(A ∧ B) ∧ C = A ∧ (B ∧ C)
(A + B) + C = A + (B + C)
This law simply switches up the bracket from the a and
b expression to the b and c expression since brackets
make an impact when doing Boolean algebra.
- Tip 1: DON’T flip the order of the inputs
- Tip 2: If the operations separating the 3 inputs
aren’t the exact same, DO NOT APPLY THE
ASSOCIATIVE LAW.
A ∧ (B + C) = (A ∧ B) + (B ∧ C)
A + (B ∧ C) = (A + B) ∧ (B + C)
This law takes the expression with 3 seperate inputs
connected by opposite operators and expands them by
multiplying the input outside the bracket with the inputs
inside the bracket.
- Tip 1: When this operation has been done, make
sure that the operation outside the brackets
initially is joined with all bracketed inputs and
connect them with the outside expression. Then
join the bracketed expressions together with the
operation inside the bracket.
Associative Laws

22. De Morgan’s Laws:
This law separates the not inputs for 2
inputs and this law also switches up
the operation being done to the inputs.
- Tip 1: If you see a not operation
inside the input and not the
expression, do the law and then
remove the not from that specific
input.
It acts like the
communicative law
except it uses the
inverse values of an
input and it swaps the
operation and not the
inputs.

23. The Theorems

24. Duality
Theore
m
This law gets a long expression of inputs
and flips the operators separating each
input.
- Tip 1: With a string of continuous or
inputs, when you then switch an or
operator to an and, place brackets
between the whole string of or
operators so you can identify where
the continuous string is occuring.
A ∧ B ∧ C + A’∧ B ∧ C =
(A + B + C) ∧ (A’+ B + C)
Consensus
Theorem
A ∧ B + A’ ∧ C + B ∧ C = A ∧ B +
A’ ∧ C
(A+B) ∧ (A’ + C) ∧ (B + C) = (A +
B) ∧ (A’ + C)
This law eliminates repeating
inputs from the expression (2 B
inputs to 1) and takes the
expressions without repeated
inputs and uses this as the output
using this law.

25. Consensus
Theorem Pt.2
- Tip 1: For the repeated inputs being
in one expression, remove the entire
sub expression from the greater
expression.
- Tip 2: This works with inverted inputs
too.
- Tip 3: If you can’t find the repeated
values in the same expression to
remove, then try applying a different
law first.
This theorem eliminates all
duplicates within an
expression by taking out
all expressions with solely
duplicate values in them.

26. Transposition Theorem
(A + B) ∧ (A’ + C) = A ∧ C + A’ ∧ B
This law swaps the operations in the
expression and this law can be done with
inverted inputs as well.
- Tip 1: Do this law after applying the
consensus law to the expression.
- Tip 2: When applying this law,
connect with the new operator the
values from the or operators
separated by the and operator and
make the new expression with inputs
from both sides.
- Tip 3: For all operations with the or
operator connecting the 2 inputs
between an and operator and the
same expression type, add brackets
to the or operation to make it easier
to find.
What’s being swapped is both the operator
and the 2 values in the brackets aren’t next
to each other in this process.
(A ∧ B + A’ ∧ C) = (A + C) ∧ (A’ + B)

27. Solving Boolean
Algebra Problems

28. - Solving Boolean Algebra Problems Happens Usually in 2 steps being finding
the algebraic expression for a problem first and then simplifying the
expression to make it easier to understand and to then be able to make the
logic gates out of the simplified answer.
A B C O
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
Problem 1
- Find and simplify the Boolean expression from the
truth table provided with the inputs provided as
well as the outputs provided.
- Tip 1: These problems can also even be given as
a statement for you to find the truth table before
doing this process.

29. Making Logic Gates Using Boolean
Algebra Expressions
- The final step needed to get a good understanding of Boolean algebra and
how to compute it is to make logic gates using the algebraic expression found
using Boolean algebra.
- Tip 1: Normally the logic gates are formed using the AND, OR & NOT gates.
- Tip 2: When making logic gates from Boolean algebra, using this example (A
+ B + C), start making the logic gates from right to left and you go from the
bottom to the top of the page.

30. Try Some Problems For Yourself
- Attempt to solve these 2 questions
1. Using this bit of Boolean algebra (A’B’C’ + A’B’C +
AB’C’ + ABC’) , simplify it and find its corresponding
logic gate
2. Using the provided truth table, find the simplified
algebraic expression and then find its corresponding
logic gate.
- MAKE SURE that you DO NOT keep going with this
video until you have found your answers to these
questions
A B C O
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1

31. Thanks For
Watching

32. Bibliography For Pictures and Ideas
https://www.vecteezy.com/vector-art/1736156-abstract-technology-futuristic-concept-of-blue-electronic-circuit
https://www.vulture.com/2021/12/the-matrix-red-pill-internet-delusional-drug.html
https://www.pond5.com/search?kw=animated-binary-numbers&media=footage
https://www.templeandwebster.com.au/Bowie-Freestanding-Mirror-TMPL3081.html
https://stock.adobe.com/au/search?k=cornerstone&load_type=tagged%20keyword&prev_url=detail
https://en.cryptonomist.ch/2022/09/05/google-play-store-truth-social-2/
https://heartofashepherd.com/2020/11/11/the-enemy-of-my-enemy-is-my-friend-luke-23-john-18-19/
https://clonetrooper.fandom.com/wiki/Clone_trooper_commander
https://www.starwars.com/databank/clone-troopers
https://www.pxfuel.com/en/desktop-wallpaper-ultrr
https://en.wikipedia.org/wiki/Augustus_De_Morgan
https://www.nature.com/articles/nrmicro.2016.98