Logical Equivalence – Domination Laws

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In the previous article, you learnt about Identity law which is an equivalence. Similarly, the domination law are another equivalence that you are going to learn in this article.

Also, this equivalence need proof which is the main purpose of this document. The prove is in the form of truth table for domination laws.

What is domination law?

The domination laws are:

\begin{aligned} &P \vee T \equiv T    \hspace{1cm} ( 1)\\ \\
&P \wedge F \equiv F  \hspace{1cm} ( 2)
\end{aligned}

First Domination Law

In the first domination law, result of P \vee True is always true. If P is a variable that stands for “I am reading” and T stands for a universal truth like “Human beings can learn“. The equivalence translates to
I am reading, or Human beings can learn

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which is equivalent to saying

I am reading, which is true, but Human beings can learn“.

This is the case when truth value of P is true.

Consider the case when P is false. Then equivalence translates to
I am not reading, or Human beings can learn“.
Now because of disjunction (\vee), if any statement is true, the compound preposition is true. Therefore, the first equivalence is valid.

Second Domination Law

Let P be the statement “I am reading” and F is a universal false statement “I can read 100 books in 5 minutes“. If P is true, the second statement P \wedge False becomes false because it is impossible to say, “I am reading, and I can read 100 books in 5 minutes”.

Therefore, the second equivalences are valid.

Suppose P is false, them both P \wedge False gives a false truth value and we have false on both sides of the equivalence. The equivalence holds.

In the first equivalence, the dominating value is true and in the second equivalence the dominating value is false, hence the law is called domination law.

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Truth Table of Domination Laws

To prove that equivalence is true for all inputs of P, we will construct the truth table for domination laws. The truth table for domination laws will have two rows because there is only one variable, which is P.

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Rows = 2^1 = 2

The column values for the truth table are P, T , F , P \vee T , and P \wedge F.

PTFP \vee TP \wedge F
TTFTF
TTFTF
FTFTF
FTFTF

The results of truth table show that P \vee T is a tautology and P \wedge F is contradiction. Therefore, the domination law is an equivalence and valid.

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