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DEMONSTRATE PROOFS
DEMONSTRATE PROOFS
At the end of this lesson, students will be able to:
Define theorem and proofs.
Identify the logical equivalence rules.
Use the rules of inference to validate arguments.
Utilize the rules of inference.
Show the proofs using rules of inference.
Introduction
Introduction
Theorem is a statement that can be shown to be true. Theorems are sometimes called propositions, facts or results.
Proof is the act of demonstrating that a theorem is true with a sequence of statements that forms an argument.
RABBIT
RABBIT
OR
OR
DUCK???
DUCK???
Use Rules of Inference to Validate Arguments
Use Rules of Inference to Validate Arguments
An argument is a sequence of statements.
To say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premise if the resulting premises are all true,
then the conclusion is also true.
Rules of Inference
Rules of Inference
A logical form or guide consisting of premises (or hypotheses) and draws a conclusion.
Simple arguments can be used as building blocks to construct more complicated valid arguments.
An argument is valid when the conclusion logically follows from the truth values of all the premises.
Each valid logical inference rule corresponds to an implication that is a tautology.
Modus Ponens (Mode of Affirming)
Consider the following argument form:
If p then q.
p.
Therefore q.
The fact that this argument is valid is called Modus Ponens
Example:
Statement-1: If the last digit of this number is 0, then this number can be divided by 10.
Statement-2: The last digit of this number is 0.
Conclusion: Therefore, this number can be divided by 10.
Modus Tollens (Mode of Denying)
Consider the following argument form:
If p then q.
Not q.
Therefore not p.
Example:
Statement-1: If Germany is located in South East Asia then Germany will join SEA Games.
Statement-2: Germany will not join SEA Games.
Conclusion: Therefore, Germany is not located in South East Asia.
Hypothetical Syllogism
Chain of if – then statements.
From the fact that one statement implies a second and the second implies a third, then we can conclude that the first statement implies the third.
If p then q
If q then r
Therefore, if p then r.
Example:
Statement-1: If you have my house key then you can unlock my house. P→Q
Statement-2: If you can unlock my house then you can take my money. Q→R
Conclusion: Therefore, if you have my house key then you can take my money. P→R
Disjunctive Syllogism
This argument says that when you have only two possibilities and you can rule one out, the other one must be the case.
Example:
Statement-1: Today is Sunday or Monday. P ∨ Q
Statement-2: Today is not Sunday. ¬P
Conclusion: Therefore, today is Monday. Q
Disjunctive Addition
Making generalization
If p is true then, p or q must be true.
Example:
Statement: I have vanilla flavored ice-cream. P
Conclusion: Therefore, I have vanilla or chocolate flavored ice-cream. P ∨ Q
Quick Exercise 1
Quick Exercise 1
What rule of inference is used in each of the following arguments?
John likes apple pies. Therefore, John likes apple pies or ice cream.
If it snows, then the roads are closed; it snows. Therefore, the roads are closed.
If it snows, then the roads are closed; the roads are not closed. Therefore, it does not snow.
To go to Hawaii, one must fly or take a boat; there is no seat on any flight to Hawaii this year. Therefore, one must take a boat to go to Hawaii this year.
Quick Exercise 2
Quick Exercise 2
Express the following arguments using the suitable symbols. What rules of inference are used in each of them?
If the sun is shining then Mary is at the beach; the sun is shining. Therefore Mary is at the beach.
Either the sun is shining or raining; the sun is not shining. Therefore it is raining.
If it is raining, then the roads are jammed; the roads are not jammed. Therefore, it does not rain.
Azman likes to read books. Therefore, Azman likes to read books or play football.
If I try hard and I have talent, then I will become a musician. If I become a musician, then I will be happy. Therefore, if I try hard and I have talent then I will be happy
WEEK 3 ACTIVITY
WEEK 3 ACTIVITY
Kindly complete the activity for Week 3 lesson as follows:
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