The document provides instructions for demonstrating DeMorgan's Laws using Venn diagrams. It defines two sets, A and B, representing types of birds. It then walks through 6 steps to prove both of DeMorgan's Laws by drawing a series of Venn diagrams showing the elements, complements, unions, and intersections of sets A and B.

1. MATH203-1403A-02
Applications of Discrete
Mathematics
Phase 1 / Individual
Project
Mark L. Simon II
Instructor: Doris Schantz
July 08, 2014

2. Instructions
Demonstrate DeMorgan’s Laws using a Venn diagram.
1. Draw a Venn diagram showing the elements of sets A, B, and the universe for
all 4 regions.
2. Draw a second diagram showing only the elements of the complement of set
A.
3. Draw a third diagram showing only the elements of the complement of set B.
4. Draw a fourth diagram showing the union of steps 2 and 3 for the first
DeMorgan's law or the intersection of steps 2 and 3 for the second DeMorgan's
law.
5. Draw a fifth diagram showing the elements of the intersection of sets A and B
for the first DeMorgan's law or the union of sets A and B for the second
DeMorgan's law.
6. Finally, draw the last diagram showing the complement of step 5. Compare
the results from step 4 against those in step 6 to prove both DeMorgan's laws. In
the absence of data elements, you can use 4 different colors to clearly indicate
the regions of the universe and follow all of the steps.

3. Instructions
Demonstrate DeMorgan’s Laws using a Venn diagram.
1. Draw a Venn diagram showing the elements of sets A, B,
and the universe for all 4 regions.
SET – A = Birds with feathers that can fly {eagle, sparrow, robin,
hawk, owl , chicken, road runner}
SET – B = Birds with feathers that cannot fly {kiwi, chicken,
black footed penguin, emu, ostrich, chicken, road runner}
A u B = {eagle, sparrow, robin, hawk, owl , starling , kiwi,
chicken, black footed penguin, emu, ostrich, chicken, road
runner}

4. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Black Footed Penguin , Emu,
Ostrich
U
A B
Chicken,
Road runner
(Region 1)
(Region 2) (Region 3)
(Region 4)

5. Instructions
2. Draw a second diagram showing only the elements of the
complement
of set A.
Compliment of A = Ac
Yellow - is designated as the compliment area.
NOTE: The Black Footed Penguin – The African Black Footed
Penguin has neither the ability to fly and is not considered to have real
feather-like qualities. The feathers have adapted to a feather-like
dense fir for cooling and heating for the area and has mass quantities
of oil within the pelt. Therefore, it will be considered a “Composite for
certain slides.

6. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Penguin, Emu, Ostrich
U
A B
Chicken,
Road runner
Black Footed
Penguin
/composite
Compliment of A = Ac

7. Instructions
3. Draw a third diagram showing only the elements of the
complement of set B.
Compliment of B = Bc
Yellow - is designated as the compliment area.

8. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Penguin, Emu, Ostrich
U
A B
Chicken,
Road runner
Black Footed
Penguin
/composite
Compliment of B = Bc

9. Instructions
4. Draw a fourth diagram showing the union of steps 2 and 3 for
the first De Morgan's law or the intersection of steps 2 and 3 for
the second De Morgan's law.
Complement of A U B = Birds with feathers that can fly and
can’t fly
In Demorgan’s Law of union compliment of the diagram on
the next image is designated as light blue.
Intersection of Ac and Bc = Ac ∩ Bc
In Demorgan’s Law of Intersect, which is the second slide from
this slide, depicts all areas where intersection takes place with
complement. The complement is shown as the featherless /
flightless Black Footed Penguin. However, it is noted that the
Road Runner and Chicken do have feathers, but choose not to
fly even though they are capable of flight for short periods.

10. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Penguin, Emu, Ostrich
U
A B
Chicken,
Road runner
Black Footed
Penguin
/Composite
Complement of A U B = (A U B)c

11. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Penguin, Emu, Ostrich
U
A B
Chicken,
Road
runner
Black Footed Penguin
/Composite
Chicken,
Road
runner
Eagle, Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin, Emu,
Ostrich
Chicken,
Road
runner
Intersection of Ac and Bc = Ac ∩ Bc
Black Footed
Penguin
/Composite
Eagle, Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin, Emu,
Ostrich
Chicken,
Road
runner
Black Footed
Penguin
/Composite
A B A B

12. Instructions
5. Draw a fifth diagram showing the elements of the intersection
of sets A and B for the first De Morgan's law or the union of sets
A and B for the second De Morgan's law. I will show both.
Complement of Ac ∩ Bc = (A U B)c
Complement of Ac 𝐔 Bc = (A ∩ B)c
. Those items not in the set are designated in light blue.
SET – A = Birds with feathers that can fly {eagle, sparrow, robin,
hawk, owl , chicken, road runner}
SET – B = Birds with feathers that cannot fly {kiwi, chicken,
black footed penguin, emu, ostrich, chicken, road runner}
A U B = {eagle, sparrow, robin, hawk, owl , starling , kiwi,
chicken, black footed penguin, emu, ostrich, chicken, road
runner}

13. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Emu, Ostrich
U A B
Black Footed Penguin
/Composite
Union of Ac ∩ Bc = (A U B)c
Chicken,
Road
Runner
Eagle,
Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin,
Emu, Ostrich
Black Footed
Penguin
/Composite
A B
Eagle,
Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin,
Emu, Ostrich
Chicken,
Road
runner
Chicken,
Road
runner
Black Footed
Penguin
/Composite
A
B
The (A ∩ B)c shows
the connection
between the two
ovals. However, the
final outcome still
encompasses the two
ovals in the De
Morgan Law leaving
only the black footed
penguin as the
complement. As
suggested on next
slide.

14. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Penguin, Emu, Ostrich
U
A B
Black Footed Penguin
/Composite
Eagle, Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin, Emu,
Ostrich
Chicken,
Road
runner
Intersection of Ac U Bc = (A ∩ B)c OR Union of Ac ∩ Bc = (A U B)c OR Both
Black Footed
Penguin
/Composite
Eagle, Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin, Emu,
Ostrich
Chicken,
Road
runner
Black Footed
Penguin
/Composite
A
B A B
Chicken,
Road
runner
The outcome is the same
for both De Morgan Laws
with the end result being
the black footed Penguin
being the complement.

15. Instructions
.
6. Finally, draw the last diagram showing the complement of step 5.
Compare the results from step 4 against those in step 6 to prove both
De Morgan's laws. In the absence of data elements, you can use 4
different colors to clearly indicate the regions of the universe and follow
all of the steps.
Within these slides I have shown that both De Morgan Laws
Complement of Ac ∩ Bc = (A U B)c
Complement of Ac 𝐔 Bc = (A ∩ B)c
Are true. One law works from the outside in and the other law works
form the inside out with the same result in the end.

16. Eagle, Sparrow, Robin,
Hawk, Owl
Kiwi, Emu, Ostrich
U
Complement of
Ac 𝐔 Bc = (A ∩ B)c
Eagle,
Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin,
Emu, Ostrich
Black Footed
Penguin
/Composite
A B
Eagle,
Sparrow,
Robin,
Hawk, Owl
Kiwi, Penguin,
Emu, Ostrich
Chicken,
Road
runner
Chicken,
Road
runner
Black Footed
Penguin
/Composite
A
B
U U
Black Footed
Penguin
/Composite
A
Ac Bc
Intersect A and B compliment should be this.
Chicken,
Road
Runner
B

17. Instructions / Part II
Define two propositions (simple statements that can be
either true or false). Give a real-world example of 2
propositions that r and s can represent. Call them r and s.
Create a truth table that shows all values of the following:

18. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
r Set - r Is a basic compilation of values within
the r-set. In this case “ I am very cold”
s Set -s Is a basic compilation of values within
the
s - set. In this case “ I need to start a
fire”.
¬ r Negation of set - r This refers to “It is
not the case that”
It “is not” the case that set – r “I am
very cold” is true or false. When
applied to a table this set can be either
true or false.
¬ s Negation of set – s This refers to “It
is not the case that”
It “is not” the case that set – r “I need
to start a fire” is true or false. When
applied to a table this set can be either
true or false.
r ^ s (^) stands for the conjunction of “r”
AND “s”
This is considered a truth value and
determines weather the values are true
or false. It also helps decide the
outcome or conclusion
CONTINUED – NEXT SLIDE

19. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
r Set - r Is a basic proposition of value within the
r-set. In this case “ I am very cold”
s Set -s Is a basic proposition of value within the
s - set. In this case “ I need to start a
fire”.
Set R = I am really cold
Set S = I need to start a fire
These are considered the propositions we will use when inputting the functions of ^ =and,
v = or, and the codes ¬ or ~ = “it is not the case that” are placed within the function. They
define the hypothesis of the outcome.
R
T
F
S
T
F

20. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
¬ r Negation of set - r This refers to “It is
not the case that”
It “is not” the case that set – r “I am
very cold” is true or false. When
applied to a table this set can be either
true or false using the “and” and “or”.
¬ s Negation of set – s This refers to “It
is not the case that”
It “is not” the case that set – s “I need
to start a fire” is true or false. When
applied to a table this set can be either
true or false using the “and” and “or”.
R ¬R
T F
F T
S ¬S
T F
F T
Each table shows the negation of Set ~R and Set ~ S. The negation of this is
Set – S and Set – R. The opposite truth or fallacy exists way function the
proposition against itself.

21. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
r ^ s (^) stands for the
conjunction of “r” AND
“s”
This table runs simple input/output to come to a comclusion as follows:
1. It is true that I am cold and it’s true I need a fire to get warm (True)
2. It is true that I am cold and I don’t need to start a fire as I am inside a
tent. (False)
3. It’s false that I am cold and I do need to start a fire. (True)
4. It’s false that I am cold and it’s false that I need to start a fire. Don’t
know how to start a fire. (false)
R S R ^ S
T T T
T F F
F T F
F F F

22. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
¬ r v s “It is not the case”
that “I am cold OR I
need to start a fire.
Negation is stated.
This can go one of four ways
1. It is true that I am not cold and it’s true I need a fire to cook with.
2. It is true that I am not cold and I don’t need a fire as I am inside a tent.
3. It’s false that I am not cold and I do need to start a fire.
4. It’s false that I am not cold and it’s false that I can start a fire as I don’t
know how to start a fire.
¬R S ¬R v S
T T T
T F T
F T T
F F F

23. Propositions – These are basically described as a declarative statements added into a
function
Proposition Definition Your Answer Explained
r ^ ¬ s I am cold and “it
is not the case
that I can start a
fire
This can go one of four ways
1. It is true that I am cold or it’s not the case that I need a fire, This is false as I
am cold.
2. It is true that I am cold or it is not the case that I need to start a fire as I am
inside a tent. This is true.
3. It’s false that I am cold or it is not the case that I need to start a fire. This is true.
I’m not cold.
4. It’s true that I am cold or it’s true that I can’t start a fire as I don’t know how to
start a fire. This is true.
R ¬S R ^ ¬S
T F F
T T T
F F F
F T F