Logical Reasoning

1. Reasoning and Introduction
to Proof
Geometry

2. The main business of
mathematics is proving
theorems.
Reasoning and Introduction
to Proof
Geometry

3. BASIC TERMS AND DEFINITIONS
A definition is an exact, unambiguous explanation of the meaning of a
mathematical word or phrase.
A theorem is a statement that can be shown to be true. Theorems are also
called facts or results.
Less important theorems are sometimes called propositions.
A theorem that is helpful in the proof of other results is called a lemma. A
corollary is a theorem that can be established directly from a theorem that
has been proved.
A proof is a valid argument that establishes the truth of a theorem.
Reasoning and Introduction
to Proof
Geometry

4. The statements used in a proof can include axioms (or postulates),
which are statements we assume to be true, the premises (or
hypotheses), if any, of the theorem, and previously proven theorems.
A conjecture is a statement that is being proposed to be a true
statement. When a proof of a conjecture is found, the conjecture
becomes a theorem.
An example used to disprove statements or conjectures is called a
counterexample
Reasoning and Introduction
to Proof
Geometry
BASIC TERMS AND DEFINITIONS

5. INDUCTIVE REASONING
Uses patterns to arrive at a conclusion(conjecture).
Reasoning and Introduction
to Proof
Geometry

6. DEDUCTIVE REASONING
• Uses facts, rules, definitions, or properties to arrive at
a conclusion.
Reasoning and Introduction
to Proof
Geometry

7. The Law of Detachment (Modus Ponens)
Consider the following set of statements that illustrate
the first law.
a. If you are an 18-year old Filipino citizen, then you
can vote.
b. Pete is an 18-year old Filipino citizen
c. Therefore, Pete can vote.
Reasoning and Introduction
to Proof
Geometry

8. Reasoning and Introduction
to Proof
Geometry
In symbols, if p represents the hypothesis of the statement and q, the conclusion,
then the Law of Detachment is usually given in the following pattern
If p is true, then q is true.
And p is true
Therefore, q is true (for the given case A)
Or
If p, then q
p
Therefore q

9. Reasoning and Introduction
to Proof
Geometry
The Law of Detachment (Modus Ponens)
•Identify the hypothesis
•If another statement has the same hypothesis
•Then you can make a conclusion

10. Reasoning and Introduction
to Proof
Geometry
Another Example:
a.If the sum of two angles is 90 degrees,
then they are complementary.
b.∠1 + ∠2 = 90
c. Therefore, ∠1 and ∠2 are
complementary

11. Reasoning and Introduction
to Proof
Geometry
As one more example of the use of modus
ponens, consider the statements.
a. If a person has a license, then he is allowed to
drive
b. Manuel has a license
c. Therefore, Manuel is allowed to drive.

12. Reasoning and Introduction
to Proof
Geometry
Example:
a. If a student is in Third Year, he takes
chemistry
b. Luis is taking chemistry
c. Therefore, Luis is in Third Year

13. Reasoning and Introduction
to Proof
Geometry
The Law of Syllogism

14. Reasoning and Introduction
to Proof
Geometry
The Law of Syllogism
•You can make a conclusion if the hypothesis and
conclusion from two conditional statements are the
same.

15. Reasoning and Introduction
to Proof
Geometry
Example
a. If I get cut, then I bleed.
b. If I bleed, then I need a bandage
c. Therefore, if I get cut, I need a bandage

16. Reasoning and Introduction
to Proof
Geometry
Example
a. If a figure is a square, then it has equal sides and
angles.
b. If a figure has equal sides and angles, then it is a
regular polygon.
c. Therefore, if a figure is a square, then it is a regular
polygon

17. Reasoning and Introduction
to Proof
Geometry
If the two statements are written in symbols, the
pattern for the syllogism is:
If p, then q.
If q, then r.
Therefore, if p, then r.

18. Reasoning and Introduction
to Proof
Geometry
Example:
a. If you like pizza with everything, then you’ll
like Jimmy’s Pizza.
b. If you like Jimmy’s Pizza, then you are a pizza
connoisseur
c. If you like pizza with everything, then you are
a pizza connoisseur.

19. Reasoning and Introduction
to Proof
Geometry
The Law of Indirect Reasoning
(Modus Tollens)
The pattern for Modus Tollens is:
If p, then q
Not q
Therefore, not p

20. Direct Proof
To prove statements using the direct proof, begin by
assuming that the hypothesis is true as the first
statement of the proof. The last statement, of course,
must be the conclusion. Between the first and last
statement, we form logical statements, using the
hypothesis, definitions, axioms (or postulates), and/or
theorems that has already been proved. Sometimes we
have to examine multiple cases before showing the
statement is true in all possible scenarios.
Reasoning and Introduction
to Proof
Geometry

21. Exercises: Use the method of direct proof to prove the
following statements.
• If a is an odd integer, then 𝑎2
+ 3𝑎 + 5 is odd
• Suppose x,y ∈Z. If x is even, then xy is even.
Reasoning and Introduction
to Proof
Geometry