Here are the answers to your Venn diagram questions: 1. A (whole numbers are integers) 2. Ø (no negative numbers are integers) 3. Ø (no integers are in C) 4. U (the union of integers and whole numbers is all integers) 5. U (the union of integers and negative numbers is all integers) 6. U (the union of integers and C is all integers) 7. Ø (no numbers are both whole and negative) 8. C (the intersection of whole numbers and C is the elements they have in common) 9. B (the union of negative numbers and C is all their elements)

1. MEMORARE
Remember, O most gracious Virgin Mary, that
never was it known that anyone who fled to thy
protection, implored thy help, or sought thine
intercession was left unaided.
Inspired by this confidence, I fly unto thee, O
Virgin of virgins, my mother; to thee do I come,
before thee I stand, sinful and sorrowful. O
Mother of the Word Incarnate, despise not my
petitions, but in thy mercy hear and answer me.
AMEN

2. 1.1
Sets, Statements,
and Reasoning

3. Set
A set is any collection of objects which
are known as elements of the set.
A = { counting numbers}
A is the set of counting numbers
B = { 2, 4, 6, 8, }
B is the set of elements 2, 4, 6, and 8

4. A = { counting numbers}
B = { 2, 4, 6, 8, }
∴B is a subset of A because
each element in B is also in A.
In symbol, B ⊆ A.

5. U = { integers}
A = { whole numbers}
B = { negative numbers}
∴B is a subset of A
because each element in B
is also in A.

6. In Geometry, geometric
figures such as lines and
angles are actually sets
of points.

7. Statement
A statement is a set of words and
symbols that collectively make a claim
that can be classified as TRUE or FALSE.
1. 10 > 17
2. The skin is the largest organ
of the human body.
3. Stop singing.
4. 7 + 19 = 26

8. Which sentences are statements?
1 . Where do you live?
2. 4 + 7 ≠ 5
3 . Quezon was the Philippine
president.
4 . X + 9 = 12 when X = 2

9. Variable
A variable is a letter representing a
number.
Some statements contain one or more
variables.
1. x + 5 = 6
2. a + b = 12

10. Open statement or open sentence
It is a statement that can be classified
as true or false depending on the
replacement value of x.
Find the replacement set.
1. x + 5 = 6
2. 7 – 2x = –39

11. Negation (∼ )
The negation of a given statement
P is a claim opposite the original
statement.
The negation of a true statement is
false, and vice versa.
If P is a statement, its negation is ∼P.

12. Give the negation of each
statement.
1. 11 + 3 = 16
2. 18 > 4
3. ∠C is a right angle.
4. I like vegetables.
5. All jokes are funny.

13. Give the negation of each
statement.
• Seoul is the Capital of South Korea.
• Christopher Columbus crossed the Atlantic
Ocean.
• All jokes are funny.
• No one likes me 
• Geometry is not my favorite subject.

14. Compound Statement
It is a statement formed by combining
other statements ( P and Q) used a
building blocks.
Conjunction: P and Q
The conjunction of P and Q
Disjunction: P or Q
The disjunction of P and Q

15. Compound Statement
Conjunction: P and Q
P: 3 + 8 = 11.
Q: LSGH is located in San Juan City.
P and Q
3 + 8 = 11 and LSGH is located in San
Juan City.

16. Compound Statements:
1. 3 + 8 = 11 and LSGH is located in
San Juan City.
2. 3 + 8 = 11 and LSGH is located in
Mandaluyong City.
3. 3 + 8 = 38 and LSGH is located in
Mandaluyong City.
4. 3 + 8 = 38 and LSGH is located
inSan Juan City.

17. Compound Statement
Disjunction: P or Q
P: 3 + 8 = 11.
Q: LSGH is located in San Juan City.
P or Q
3 + 8 = 11 or LSGH is located in San
Juan City.

18. Compound Statements:
1. 3 + 8 = 11 or LSGH is located in
San Juan City.
2. 3 + 8 = 11 or LSGH is located in
Mandaluyong City.
3. 3 + 8 = 38 or LSGH is located in
Mandaluyong City.
4. 3 + 8 = 38 or LSGH is located
inSan Juan City.

19. The Table for Conjunction The Table for Disjunction
--A conjunction is TRUE only when P and Q are
BOTH TRUE.
--A disjunction is FALSE only when P and Q are
BOTH FALSE.

20. Conditional Statement
(or implication)
“If P, then Q.”
If an angle is a right
angle, then it measures
90 .
o

21. If an angle is a right
angle, then it measures
90 .
o
All right angles measure
90 .
o
(also a conditional statement)

22. Conditional Statements: ( T or F )
1. All right angles measure 90 . o
2. An angle whose measure is 90o
is a right angle.
3. If angles are vertical, then they
are congruent.
4. If I study hard, I will get high
grades.
5. Love me, and I’ll love you back.

23. Classify each statement as simple,
conditional, a conjunction or a
disjunction.
1. If Alice plays, the volleyball team will win.
2. The first-place trophy is beautiful.
3. An integer is odd or it is even.
4. I enrolled in Gold’s Gym and Fitness First.
5. Charlie’s neat and Charlie’s sweet.
6. You will be in trouble if you don’t change
your ways.

24. State the hypothesis and
conclusion.
1. If you go to the game, then you will have a
great time.
2. If two chords of a circle have equal lengths,
then the arcs of the chords are congruent.
3. Vertical angles are congruent when two
lines intersect.
4. Corresponding angles are congruent if two
parallel lines are cut by a transversal.

25. Reasoning
Reason is a term that refers to the capacity human beings have
to make sense of things, to establish and verify facts, and to
change or justify practices, institutions, and beliefs. It is closely
associated with human activities as philosophy, science,
language, mathematics, and art. The concept of reason is
sometimes referred to as rationality and sometimes as
discursive reason, in opposition to intuitive reason.
Reason or "reasoning" is associated with thinking, cognition,
and intellect. Reason, like habit or intuition, is one of the ways
by which thinking comes from one idea to a related idea. For
example, it is the means by which rational beings understand
themselves to think about cause and effect, truth and falsehood,
and what is good or bad.

26. Reasoning
It is a process based on experience and principles that
allow one to arrive at a conclusion.
1. INTUITION is an inspiration leading to the statement
of a theory. It is a sudden insight allowing one to make
a statement without applying any formal reasoning.
2. INDUCTION an organized effort to validate a theory.
(PARTICULAR TO GENERAL)
3. DEDUCTION it is a formal argument that proves the
tested theory.
(GENERAL TO PARTICULAR)

27. Use induction to state a
conclusion.
Several movies directed by Lawrence
Garrison have won Awards, and many
others have received nominations.
His latest work, A Prisoner of Society,
is to be released next week.
Conclusion?

28. While searching for a classroom, Tom
stopped at an instructor’s office to
ask directions. On the office
bookshelves are books titled
Intermediate algebra, Calculus,
Modern Geometry and Linear
Differential Equations. Conclusion?

29. At a friend’s house, you see
several food items including
apples, pears, grapes, and
bananas.
Conclusion?

30. On Monday, Matt says to you, “Andy hit his
little sister at school today”. On Tuesday, Matt
informs you, “Andrew threw his Math book
into the wastebasket during class.” On
Wednesday, Matt tells you,”Because Andy was
throwing peas in the school cafeteria, he was
sent to the principal’s office.”
Conclusion?

31. Use deduction to state a
conclusion, if possible.
If the sum of two angles is 90 degrees,
then these angles are
“complementary.”
Angle 1 measures 27 degrees and
Angle 2 measures 63 degrees.
Conclusion?

33. Law of Detachment
1. If P, then Q.
premises
2. P
C. ∴ Q conclusion
1. If a man lives in Davao, then he
lives in the Philippines.
2. Dan lives in Davao.
C. ∴Dan lives in the Phil.

34. Law of Detachment
1. If a man lives in Davao, then he
lives in the Philippines.
2. Dan lives in the Philippines.
C. ∴Dan lives Davao.
Valid Argument Invalid Argument
1. If P, then Q. 1. If P, then Q.
2. P 2. Q
C. ∴ Q C. ∴ P

35. Law of Detachment
1.If an angle is a right angle, then it
measures 90o.
2. If you study hard, then you’ll get
high grades.
3. If an animal is a cat, then it makes a
meow sound.

36. Venn Diagram
A Geometric figure representing
sets of objects. It is created by
John Venn , an Englishman.
Each set is represented by a
closed figure such as circle or
rectangle.

37. Venn Diagram
1. If a man lives in 1. If a man lives in
Davao, then he lives Davao, then he lives in
in the Philippines. the Philippines.
2. Ken lives in Davao. 2. Ken lives in the
C. ∴Ken lives in the Philippines.
Phil. C. ∴Ken lives in Davao
IS
C IS
P Q K?
C
etc D etc P
K

38. Venn Diagram
1. If a man lives in Sucat, then he lives
in Parañaque.
2.If an angle is a right angle, then it
measures 90o.
3. If two angles are vertical, then they
are congruent.

39. Venn Diagram
Conjunction (and)
Intersection ( ∩ )
-set of elements common to both
sets
Disjunction (or)
Union (∪)
-all the elements from both sets

40. Venn Diagram
U = { integers}
A = { whole numbers}
B = { negative numbers}
C = { 2, 4, 6, 8, 10}
Find:
1. U ∩ A 4. U ∪ A 7. A ∩ B
2. U ∩ B 5. U ∪ B 8. A ∩ C
3. U ∩ C 6. U ∪ C 9. B ∪ C