Prove p ∧ (p → q) → q is a tautology using a truth table

[Tek1Atom]

Homework Statement

By providing a truth table, show that the following logic statement is a tautology:

p ∧ (p → q) → qAny help will be much appreciated.

Thank You

 

[Mark44]

I would set up a truth table with columns for p, q, and p ==> q. You know how many rows you need, right?

 

[Tek1Atom]

p | q | P → q
-------------
T | T | T
T | F | F
F | T | T
F | F | T

Thank you Mark44 but I don't think this is the correct answer/method as the output has to be true in all cases in order for it to be labeled a tautology. Any other suggestions?

 

[D H]

You are missing at least one column, multiple columns if you want to break things down.

 

[Tek1Atom]

D H could you give me an example please as I am new to logic...

 

[D H]

You have the truth table for implication. p→q obviously is not a tautology. You weren't asked to show that. You were asked to show that p∧(p→q)→q is a tautology. Your truth table needs to end with a p∧(p→q)→q column on the right and with all four values in this column being T.

 

[Tek1Atom]

Thank you DH, that was very helpful. My final question is, does it matter which way we write down the truth table?

e.g.

which one is correct?

p | q | P → q | p∧(p→q)→q
---------------------------
T | T | T | T
T | F | F | T
F | T | T | T
F | F | T | T

OR

p | q | P → q | p∧(p→q)→q
---------------------------
F | F | T | T
F | T | T | T
T | F | F | T
T | T | T | T

 

[D H]

You aren't going to get points for that. At least I hope you aren't. There is a thing called showing your work.

 

[Mark44]

Thank you DH, that was very helpful. My final question is, does it matter which way we write down the truth table?

e.g.

which one is correct?

p | q | P → q | p∧(p→q)→q
---------------------------
T | T | T | T
T | F | F | T
F | T | T | T
F | F | T | T

OR

p | q | P → q | p∧(p→q)→q
---------------------------
F | F | T | T
F | T | T | T
T | F | F | T
T | T | T | T

To answer your question about the order, it doesn't really matter, but these are usually presented with the top row being T T ... and the bottom row being F F ...

 

[Tek1Atom]

Thank You Mark44. You have been great!