Proof: τ(p∨q)=1-τ(¬p∧¬q)=

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Thread starter skaeno
Start date Sep 7, 2013
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Logic

In summary, the expression τ(p∨q)=1-τ(¬p∧¬q) represents the logical equivalence between a disjunction and a negation of a conjunction. It is a proof that shows these two statements have the same truth value. An example can be seen with the statements "It is raining" and "I am carrying an umbrella". This expression is always true and is used in scientific research to demonstrate logical connections between statements.

Sep 7, 2013

skaeno

If τ(¬p)=1-τ(p) and τ(p∧q)=τ(p)∙τ(q) show that:
τ(p∨q)=τ(p)+τ(q)-τ(p)∙τ(q)

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Sep 7, 2013

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skaeno said:

If τ(¬p)=1-τ(p) and τ(p∧q)=τ(p)∙τ(q) show that:
τ(p∨q)=τ(p)+τ(q)-τ(p)∙τ(q)

Hi skaeno, welcome to MHB! :)

Perhaps you might consider that p∨q=¬(¬p∧¬q)?

1. What is the meaning of the expression τ(p∨q)=1-τ(¬p∧¬q)?

The expression τ(p∨q)=1-τ(¬p∧¬q) is a mathematical representation of the logical equivalence between a disjunction (p∨q) and a negation of a conjunction (¬p∧¬q). It states that the truth value (τ) of a disjunction (p∨q) is equal to 1 (true) minus the truth value of a negation of a conjunction (¬p∧¬q).

2. How does this expression relate to the concept of proof?

This expression is a logical proof that demonstrates the equivalence between a disjunction and a negation of a conjunction. It shows that these two logical statements have the same truth value, meaning they are both either true or false in the same circumstances.

3. Can you provide an example to explain this expression?

Sure, let's use the following statements: p = "It is raining" and q = "I am carrying an umbrella". In this case, p∨q would mean "It is either raining or I am carrying an umbrella" and ¬p∧¬q would mean "It is not raining and I am not carrying an umbrella". The expression τ(p∨q)=1-τ(¬p∧¬q) can be demonstrated by showing that if p is true (it is raining) and q is false (I am not carrying an umbrella), then p∨q is true and ¬p∧¬q is false, resulting in the same truth value for both expressions.

4. Is this expression always true?

Yes, this expression is a logical tautology, meaning it is always true regardless of the truth values of p and q. This can be proven by constructing a truth table for both expressions, which will show that they always have the same truth value.

5. How is this expression used in scientific research?

This expression is used in scientific research to demonstrate the logical equivalence between two statements. It can be applied in various fields, such as mathematics, computer science, and physics, to prove the validity of certain hypotheses or theories. It also allows scientists to make deductions and draw conclusions based on the logical connections between different statements.