1.1 sets, statements, and reasoning

MEMORARERemember, 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

1.1Sets, Statements, and Reasoning

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

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.

U = { integers}A = { whole numbers}B = { negative numbers}

B is a subset of A because each element in B is also in A.

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

StatementA statement is a set of words and symbols that collectively make a claim that can be classified as TRUE or FALSE.

1. 10 > 172. The skin is the largest organ of the human body.3. Stop singing.4. 7 + 19 = 26

Which sentences are statements?

1. Where do you live?2. 4 + 7 ≠ 53. Quezon was the Philippine

president.4. X + 9 = 12 when X = 2

VariableA variable is a letter representing a number.

Some statements contain one or more variables.1. x + 5 = 6

2. a + b = 12

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 = 62. 7 – 2x = –39

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.

Give the negation of each statement.1. 11 + 3 = 162. 18 > 43. C is a right angle.4. I like vegetables.5. All jokes are funny.

Give the negation of each statement.

1. Seoul is the Capital of South Korea.2. Christopher Columbus crossed the Atlantic

Ocean.3. All jokes are funny.4. No one likes me 5. Geometry is not my favorite subject.

Compound StatementIt is a statement formed by combining other statements ( P and Q) used a building blocks.

Conjunction: P and QThe conjunction of P and Q

Disjunction: P or QThe disjunction of P and Q

Compound StatementConjunction: P and QP: 3 + 8 = 11.Q: LSGH is located in San Juan City.

P and Q3 + 8 = 11 and LSGH is located in San Juan City.

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.

Compound StatementDisjunction: P or QP: 3 + 8 = 11.Q: LSGH is located in San Juan City.

P or Q3 + 8 = 11 or LSGH is located in San Juan City.

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.

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.

Conditional Statement(or implication)

“If P, then Q.”If an angle is a right angle, then it measures 90o.

If an angle is a right angle, then it measures 90o.

All right angles measure 90o. (also a conditional statement)

Conditional Statements: ( T or F )1. All right angles measure 90o. 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.

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.

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.

ReasoningReason 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.

ReasoningIt 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)

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?

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?

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

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?

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?

Law of Detachment

1. If a man lives in Davao, then he lives in the Philippines.2. Dan lives in Davao.C. Dan lives in the Phil.

1. If P, then Q.2. PC. Q

premises

conclusion

Law of Detachment1. If a man lives in Davao, then he lives in the Philippines.2. Dan lives in the Philippines.C. Dan lives Davao.

1. If P, then Q.2. PC. Q

Valid Argument

1. If P, then Q.2. QC. P

Invalid Argument

Law of Detachment1.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.

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.

Venn Diagram

1. If a man lives in Davao, then he lives in the Philippines.2. Ken lives in Davao.C. Ken lives in the Phil.

P

D

1. If a man lives in Davao, then he lives in the Philippines.2. Ken lives in the Philippines.C. Ken lives in Davao

Q

PK

K?

ISISC C

etcetc

Venn Diagram1. 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.

Venn DiagramConjunction (and)Intersection ( )-set of elements common to both sets

Disjunction (or)Union ()-all the elements from both sets

Venn DiagramU = { integers}A = { whole numbers}B = { negative numbers}C = { 2, 4, 6, 8, 10}Find:1. U A 4. U A 7. A B2. U B 5. U B 8. A C3. U C 6. U C 9. B C