Proving Logic Statements with Truth Tables and Laws

In summary, when it comes to proving logical equivalences in homework, there are two main methods that can be used: truth tables and algebra rules. Truth tables are more suitable when there are a low number of symbols but a high number of logical combinations, while algebra rules may be faster or easier when dealing with many symbols or simpler expressions. Ultimately, the method used will depend on the specific problem and assignment requirements.
  • #1
wdulli
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Homework Statement



Sometimes i got a question with homework if i can prove something. I got a book where the tolled me to use truthtables to see the outcome :

For example :

(notP V Q) <=> (P => Q)

P Q | notP | notP v Q | P=>Q | (notP V Q) <=> (P => Q)
0 0 | 1 | 1 | 1 | 1
0 1 | 1 | 1 | 1 | 1
1 0 | 0 | 0 | 0 | 1
1 1 | 0 | 1 | 1 | 1

Lateron in the same book it's statement is, that it's easier to use the Logic laws then written down the truthtable everytime. And gives the logic laws for example

1. Double complement
2. The morgan
3. Commutative
4 Associative
etc etc

No my question :

When do i know what to use. Do i allways start with Double complement
, the morgan , Commutative etc.

The Attempt at a Solution



Searched on the internet (Different websites) but i can't find the solution/way. I didnt put the question in this forum post because i want to know the steps to take not the answer.
 
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  • #2
You can in general use either or both methods (truth tables and algebra rules) as you see fit.

Usually you will want to use truth tables when the number of distinct symbols is low but the number of logical combinations is high (like if you have to evaluate the equivalence of two rather long expressions involving only P and Q). If there is N distinct symbols the truth table has 2N rows, so for N = 2 or 3 that is really easy.

On the other hand, if you have many symbols, or the expressions are simple, or you can recognize sub-expressions from the list of rules you know, it may be faster or easier to use the rules to prove equivalence.

Of course, you may get an assignment that ask you to use either method to prove something and then you of course have to use that method.
 

FAQ: Proving Logic Statements with Truth Tables and Laws

What is the purpose of using truth tables and laws to prove logic statements?

The purpose of using truth tables and laws is to provide a systematic and logical approach to proving the validity of logical statements. Truth tables help to analyze the different combinations of inputs and outputs, while laws provide a set of rules for manipulating and simplifying logical expressions.

What is a truth table and how is it used to prove logic statements?

A truth table is a graphical representation of all possible combinations of inputs and their corresponding outputs for a logical statement. It is used to prove logic statements by showing that the output is always true for every possible combination of inputs, thereby proving the validity of the statement.

What are the basic laws used in proving logic statements?

The basic laws used in proving logic statements are the Commutative Law, Associative Law, Distributive Law, De Morgan's Laws, Identity Law, and Negation Law. These laws provide a set of rules for manipulating and simplifying logical expressions.

What is the difference between a tautology and a contradiction in logic statements?

A tautology is a logical statement that is always true, regardless of the inputs. This can be proven using a truth table by showing that the output is always true. On the other hand, a contradiction is a logical statement that is always false, regardless of the inputs. This can be proven using a truth table by showing that the output is always false.

Can truth tables and laws be used to prove all types of logic statements?

Yes, truth tables and laws can be used to prove all types of logic statements, including propositional logic, predicate logic, and Boolean logic. However, for more complex statements, other methods such as mathematical induction or proof by contradiction may be used in addition to truth tables and laws.

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