No Tautology

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http://img222.imageshack.us/i/tautology.png/

I'm not really 100% sure, but it should be:

a) Tautology
b) Tautology Right and left of <-> need to have the same value, right? I think that's the case here.
c) No Tautology, I think. If p, then r and if q then r. contradictory statement? No Tautology then.
d) don't really see a condition here. if p, then p and q... huh? I guess it's not always true. No Tautology.
e) No Tautology. It's not always true.


There must be an easy way to do this! I can't always go through every single statement and check if those 4-5 statements in one equation are always true. Takes too much time and it's too confusing. Is there any other way to do this??



Oh yeah, it's not homework! :rolleyes:
 

Answers and Replies

  • #2


There must be an easy way to do this! I can't always go through
every single statement and check if those 4-5 statements in one equation are always true.
Takes too much time and it's too confusing. Is there any other way to do this??
Do you know what a truth table is?

Do you know how to construct a truth table?

Do you understand the reason why a truth table is used to verify a tautology?

Do you understand why the table of truth values in a logical implication is the way it is?

If you can't answer all these questions I think you need to read an elementary logic book,
online logic notes or the logic chapter of a discrete math text at least.
 
  • #3
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Unfortunately you have to make truth tables and compare the values on each column. If two columns have identical values then they are tautologies.
 
  • #4
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Unfortunately you have to make truth tables and compare the values on each column. If two columns have identical values then they are tautologies.
So, you mean like, for instance for a, I would do : p->q and p'->r / q V r ?? So 2 truth tables?
 
  • #5
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So, you mean like, for instance for a, I would do : p->q and p'->r / q V r ?? So 2 truth tables?
For a you would need one table with the truth values for:

p[tex]\rightarrow[/tex]r

[tex]\neg[/tex]p[tex]\rightarrow[/tex]r

(p[tex]\rightarrow[/tex]r)[tex]\wedge[/tex]([tex]\neg[/tex]p[tex]\rightarrow[/tex]r)

q[tex]\vee[/tex]r

(p[tex]\rightarrow[/tex]r)[tex]\wedge[/tex]([tex]\neg[/tex]p[tex]\rightarrow[/tex](q[tex]\vee[/tex]r)

It becomes 1 table with 5 columns.
 
  • #6
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Oh and don't forget the columns for p, q, r. So a table with 8 columns that is.
 
  • #7
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For a you would need one table with the truth values for:

p[tex]\rightarrow[/tex]r

[tex]\neg[/tex]p[tex]\rightarrow[/tex]r

(p[tex]\rightarrow[/tex]r)[tex]\wedge[/tex]([tex]\neg[/tex]p[tex]\rightarrow[/tex]r)

q[tex]\vee[/tex]r

(p[tex]\rightarrow[/tex]r)[tex]\wedge[/tex]([tex]\neg[/tex]p[tex]\rightarrow[/tex](q[tex]\vee[/tex]r)

It becomes 1 table with 5 columns.
But that's not the one for a, though. And you need p twice... p and p'.

So the last one contains the entire compound proposition, and it its value is F, and the other 7 have a value of F, it's a Tautology?


Wikipedia says..."Because each row of the final column shows T, the sentence in question is verified to be a tautology.". So if p, q, and r are all F and the rest is T, it's a Tautology?
 
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  • #8
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But that's not the one for a, though. And you need p twice... p and p'.

So the last one contains the entire compound proposition, and it its value is F, and the other 7 have a value of F, it's a Tautology?


Wikipedia says..."Because each row of the final column shows T, the sentence in question is verified to be a tautology.". So if p, q, and r are all F and the rest is T, it's a Tautology?
Oh yeah you're right. I was watching b I guess. But you got the idea. And yes you need the p' too.

Sorry btw, I got a bit confused. Tautology is not what I said earlier. That was a retoric tautology, something completely different. In logic tautology is just a statement that is ALWAYS true. So if all values in a column are true then it's a tautology. If all values are false then it's called a contradiction.

Tautology -> a OR a'

Contradiction -> a AND a'

In the first one we will always get true no matter if a is false or true and in the second one we will always get false, no matter if a is true or false.
 
  • #9
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Propositional logic begins with propositional variables, atomic units that represent concrete propositions. A formula consists of propositional variables connected by logical connectives in a meaningful way, so that the truth of the overall formula can be uniquely deduced from the truth or falsity of each variable. A valuation is a function that assigns each propositional variable either T (for truth) or F (for falsity). So, for example, using the propositional variables A and B, the binary connectives \lor and \land representing disjunction and conjunction, respectively, and the unary connective \lnot representing negation, the following formula can be obtained:

(A \land B) \lor (\lnot A) \lor (\lnot B).

A valuation here must assign to each of A and B either T or F. But no matter how this assignment is made, the overall formula will come out true. For if the first disjunct (A \land B) is not satisfied by a particular valuation, then one of A and B is assigned F, which will cause the corresponding later disjunct to be T.
 

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