proof demonstrate math, Exercises of Mathematics

Download proof demonstrate math and more Exercises Mathematics in PDF only on Docsity!

Basic Logic and Proofs

Topic 1

Subtopics

♥ 1.1 Derive propositional logic

♥ 1.2 Derive predicate logic

♥ 1.3 Demonstrate proofs

7 Rules of Inference (Valid)

❑ Addition

❑ Conjunction

❑ Simplification

❑ Modus Ponens

❑ Modus Tollens

❑ Disjunctive Syllogism

❑ Hypothetical Syllogism

7 Types of rules of inference

Rules of inference Name Rules of inference Name P ∴ P ∨ Q Addition P ∧ Q ∴ P Simplification P Q ∴ P ∧ Q Conjunction P P ⟶ Q ∴ Q Modus Ponens P ∨ Q ∼ P ∴ Q Disjunctive Syllogism P ⟶ Q Q ⟶ R ∴ Q Hypothetical Syllogism ∼ Q P ⟶ Q ∴ ∼ P Modus Tollens

Example 2: Determine the rules of

inference for the following arguments.

Question:

Ahmad and Ali go to school. Therefore, Ahmad goes to

school.

p : Ahmad goes to school. q : Ali goes to school.

Answer:

Premise 1 : P ∧ Q

Conclusion : ∴ P

Rule of inference: Simplification.

Example 3: Determine the rules of

inference for the following arguments.

Question: If sun is hot, then water is cold. If water is cold, then I am giggling. Therefore, if sun is hot, then I am giggling. P : Sun is hot Q : water is cold R : I am giggling Answer: Premise 1: P ⟶ Q Premise 2: Q ⟶ R Conclusion: ∴ P ⟶ R Rule of inference: Hypothetical Syllogism

Example 1

❑ If roses are red and violets are blue, then sugar is sweet and so are you. Roses are red and violets are blue. Therefore, sugar is sweet and so are you. p : Roses are red q : Violets are blue r : Sugar is sweet s : You are sweet Premise 1 : (p ∧ q) ⟶ (r ∧ s) Premise 2 : p ∧ q _______________________________ Conclusion : ∴ r ∧ s Compare to rules of inference. If does not match with any rules, thus, students need to construct table.

Solution:

Premise 1 : (P ∧ Q) ⟶ (R ∧ S) Premise 2 : P ∧ Q Conclusion : ∴ R ∧ S Answer: The arguments are valid. Premise 2 Conclusion Premise 1 Conclusion P Q R S 𝑃 ∧ 𝑄 𝑅 ∧ 𝑆 (𝑃 ∧ 𝑄) → (𝑅 ∧ 𝑆) 𝑅 ∧ 𝑆 T T T T T T T T T T T F T F F F T T F T T F F F T T F F T F F F T F T T F T T T T F T F F F T F T F F T F F T F T F F F F F T F F T T T F T T T F T T F F F T F F T F T F F T F F T F F F F T F F F T T F T T T F F T F F F T F F F F T F F T F F F F F F F T F

Solution

Premise 1 Premise 2 Conclusion Premise 3 Conclusion P Q R P → Q ~R Q → ~R ~R T T T T F F F T T F T T T T T F T F F T F T F F F T T T F T T T F F F F T F T T T T F F T T F T F F F F T T T T

Premise 1 : P

Premise 2 : P → Q

Premise 3 : Q → ~R

Conclusion : ∴ ~R

Answer: The arguments

are valid.

Example 3

Premise 1: P Premise 2: Q Premise 3: P → Q Conclusion: ∴ R Answer: The arguments are invalid. Premise 1 Premise 2 Premise 3 Conclusion P Q R P → Q R T T T T T T T F T F T F T F T T F F F F F T T T T F T F T F F F T T T F F F T F

EXERCISES:

1. Elaborate the validity of those arguments by: ❑ Someone is sick. Someone is not happy. So, someone is sick and not happy. ❑ If he loves me then he gives me flowers. He gives me flowers. Thus, he loves me. ❑ If it rains, the streets will be wet. If the streets are wet, accidents will happen. Therefore, accidents will happen if it rains. ❑ If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on. If the sailing race is held, then the trophy will be awarded. The trophy was not awarded. It concludes that it rained.