•STATEMENTS•TYPES OF REASONING

•CONDITIONAL STATEMENTS•WRITING – UP PROOFS

Introduction to Proving

Statements

These are sentences that are either true or false but not both.

Examples:p: The sun rises in the morning.q: 2.54 is an integer.r: 4 + 4 = 8s: 3 is an even integer.

Not Statements…

Your place or mine?Why is induction important?Go to the Prefect’s Office.Knock before entering!It is hot today.3 + x = 5

Compound Statements

Sentences combining two or more ideas that can be assigned a truth value. (that is, true or false)

Examples:The number 5 is not an integer.The number 4 is positive and the number 3 is

negative.If a set has n elements, then it has 2n subsets.

Compound Statements

2n + n is a prime number for infinitely many n.

Every even integer greater than 2 is a sum of two prime numbers.

x + y = y + x, for all real numbers x and y.It is not true that 3 is an even integer or 7 is

a prime.If the world is flat, then 2 + 2 = 4.

Negations ( ~ p )

Write the negation of the following statements:

p: The sun rises in the morning.

q: 2.54 is an integer.r: 4 + 4 = 8s: 3 is an even integerp ~ p

T F

F T

Negating Quantifiers

Form of Statement Form of Negation

All are Some are not

None are Some are

Some are None are

Some are not All are

Examples

All students in this class will get 100 in Geometry.

Some calculators are solar powered.

No money grows on trees.Some cows do not give milk.

Connectives

Examples are and, or, if … then, if and only if etc.

Conjunction (Λ) – two simple statements joined by the word “and”.

Disjunction (V) – two simple statements joined by the word “or”.

Example:

Let p and q be simple statements.p: The number 3 is an even integer.q: The number 7 is prime.

“ The number 3 is an even integer and the number 7 is prime.”

“ The number 3 is an even integer or the number 7 is prime.”

Consider the ff. statements:

Albay is in Bicol and 5 + 5 = 10.Albay is in Bicol and 5 + 5 = 11.Albay is in Laguna and 5 + 5 = 10.Albay is in Laguna and 5 + 5 = 12.

Which of the following is true?

For Conjunction

p q p Λ qT T TT F FF T FF F F

Consider the ff. statements:

Marist is in Marikina or 5 + 5 = 10.Marist is in Marikina or 5 + 5 = 11.Marist is in QC or 5 + 5 = 10.Marist is in QC or 5 + 5 = 12.

Which of the following is true?

For Disjunction

p q p V qT T TT F TF T TF F F

Let p and q be statements given by:

p: Triangle ABC has three sides.q: Triangle ABC has sides of the same length.

Write the following in symbols.a.Triangle ABC has 3 sides and its sides are of

the same length.b. Triangle ABC has 3 sides or its sides are of

the same length.c.Triangle ABC has 3 sides and its sides are

not of the same length.d.It is not true that triangle ABC has 3 sides

and its sides are of the same length.

p: 6 is even ; q: 9 < 5

p Λ qp V qp Λ ~ q~ p V q

Consider the statement:

If I will be elected, then I will be your voice in the administration.

When will this person becomes true in his statement?

If then

True True

True False

False True

False False

p - > q

T

F

T

T

Conditional Statements

If – then statementsFormed by the connectives if and then

If – statements are called hypotheses

Then – statements are called conclusions

p –> q

If p then qp implies qp only if qp is sufficient for qq is necessary for p

Write each statement in “if-then” form.

All intelligent students can pass mathematics.

All right angles are equal.Equal quantities multiplied by equal

quantities are equal.All triangles have three sides.An obtuse angle is greater than an

acute angle.It is colder in winter.

Determine if the following statements are true.

If Boracay is in Bicol, then 3 + 3 = 6

If Boracay is not in Bicol, the 3 + 3 = 6

If Boracay is not in Bicol, then 3 + 3 = 4

If Boracay is in Bicol the 3 + 3 = 4

Bi – Conditional Statements

“ p if and only if q”p <-> qp is necessary and sufficient for q.p is equivalent to q.It is true when p and q are both true or both

false.p q p <-> q

T T T

T F F

F T F

F F T

Check Your Understanding

If p<->q is true, what conclusion can be drawn from the statement: (~q v p) -> q ?

p q ~q ~q v p (~q v p) -> q

T T F T T

F F T T F

Converse of a statement

If the hypothesis and the conclusion in an implication are reversed, the new statement is called the converse of the given statement.

If q, then p.

Example: Statement: Every dog is an animal. Converse: Every animal is a dog.

State the converse of each statement:

If two sides of a triangle are equal, then the angles opposite those sides are equal.

The diagonals of a rectangle are equal.

When it is raining, it is colder.

Inverse of a statement

The inverse of a statement , “If p, then q.” is “If not p then not q.”

~ p -> ~ q

ExamplesIf two sides of a triangle are equal, then the

angles opposite those sides are equal.Circles having equal radii are congruent.When it is raining, it is colder.

Contrapositive of a statement

It is the converse of its inverse.~ q -> ~ p

ExamplesIf two sides of a triangle are equal, then

the angles opposite those sides are equal.Circles having equal radii are congruent.When it is raining, it is colder.

Consider the statement:

The opposite angles of a parallelogram are equal.

Give the following:ConditionalConverseInverseContrapositive

Answers

Conditional If two angles are opposite angles of a parallelogram,

then they are equal.

Converse If two angles of a parallelogram are equal, then they

are opposite angles.

Inverse If two angles of a parallelogram are not opposite

angles, then they are unequal.

Contrapositive If two angles of a parallelogram are unequal, then

they are not opposite angles.

Types of Reasoning

Reasoning by Intuition

Reasoning by AnalogyReasoning by InductionReasoning by Deduction

Reasoning by Analogy

It involves seeing similarities in different situations. If two situations are alike in certain ways, then perhaps they are also alike in others.

Examples1.Paeng, who was a good Mathematics

student in grade school, concludes that he will also be a good Mathematics student in high school.

2.The peso is to the Philippines as the dollar is to the United States.

Reasoning by Induction

It involves making a general statement based on several instances in which this statement is true.

Examples1.Twelve is an even number and is divisible

by 2. Six hundred eight is an even number and is divisible by 2. Therefore, all even numbers are divisible by 2.

Reasoning by Induction

It involves making a general statement based on several instances in which this statement is true.

Examples2. Tonichi is a third year student enrolled in

Chemistry. Don is a third year student enrolled in Chemistry. Ross is also a third year student enrolled in Chemistry. Therefore, all third year students are enrolled in Chemistry.

Reasoning by Deduction

It involves reaching a conclusion from previously accepted statements. These statements contain the conditions which must be met for the conclusion to be true.

Reasoning by Deduction

Examples1.Our house is beside a large mango tree and

at the back of a store. You have just passed a house beside a large mango tree but with no store at the back. You then conclude that it is not our house.

2.Any two right angles are congruent. Angle A and angle B are right angles. Therefore, angle A and angle B are conguent.