Tautology truth table

A proposition is a statement that has a truth value. The truth value is a representation in logical operations having two types: true and false. An example of a proposition is the statement $2 < 5$ whose truth value is true as it is a correct mathematical expression.

What is a tautology?

A tautology is a proposition that is always true, regardless of the truth values of the individual propositions that produce the result. This concept is useful in mathematics and sciences to help us derive new statements based on existing true statements. Moreover, it helps determine the accuracy of a statement that assists us in decision-making.

Truth table examples

Let's discuss some examples of tautology: one and two propositions.

Using one proposition

Let's understand tautology with an easy example involving just one proposition: $m$. In the following table, we calculate the negationIt returns result opposite to the input condition. of the proposition represented as $∼m$. Then, we take ORIt returns true if one of the input conditions is true. It returns true if all input conditions are true else false for all cases.of $m$ and $∼m$ represented as $m \space ∨ ∼m$.

Note: Here T refers to true while F refers to false.

We conclude that $m \space ∨∼m$ is a tautology as it results in all true outputs.

Two propositions

Let's dive deep into the tautology concept now involving two propositions: $m$ and $n$. We want to calculate $(m ∨ n) ∨ (∼m ∨ n)$, so we find $(m ∨ n)$ and $(∼m ∨ n)$ individually first. Then, we take the OR between them to get our desired output.

We conclude that $(m ∨ n) ∨ (∼m ∨ n)$ is a tautology as it results in all true outputs.

Conclusion

Tautologies play a vital role in various areas of science. They help us determine the accuracy and correctness of a statement to improve the quality of our systems. In addition, they can help us determine the correctness of our existing statements so we can derive new statements on their basis.

Let's test what we have learned so far.