Assignment 17

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course MTH 151

6:35 pm 04/03/13

Question: `q001. There are 9 questions in this set.

Explain why [ (p -> q) ^ (q -> r) ^ p] -> r must be true for every set of truth values for which r is true.

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Your solution:

In the book it states that to test the validity of an argument with a truth table (Step 4) states “Complete the truth table for the conditional statement formed in Step 3. If it is a tautology, then the argument is valid; otherwise, it is invalid.

In this case from my understanding r makes the entire statement true because if r was false the entire statement would be false

confidence rating #$&*: 3

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Given Solution:

[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true.

You could make a table, which would be useful in understanding the above explanation.

STUDENT COMMENT: I still don't quite grasp this. Is this the same thing as the table?

INSTRUCTOR RESPONSE: On any question where you don't understand the given solution, you should break the given explanation up into phrases and tell me what you do and do not understand about each. For example, on this problem you might break the explanation up as follows:

[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

TTT, TFT, FTT, FFT are all the truth values that have r true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

Now putting it all together: [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?)

STUDENT COMMENT: so r is the term that makes it true or false

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Self-critique Rating:ent: 3

INSTRUCTOR RESPONSE: The consequent r does by itself does not necessarily determine the truth of the statement.

If r is true, then the statement is true.

However if r is false then the statement might be true or false. If the conclusion r is false, then if the antecedent (in this case [ (p -> q) ^ (q -> r) ^ p]) is true the statement is false. However if the antecedent is false, then the statement is true, despite the fact that r is false.

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Self-critique (if necessary):

I think I have the right idea

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Self-critique Rating: OK