Show that the conditional statement is a Tautology without using truth tables

[VinnyCee]

Homework Statement

Show that $$\left[\neg\,p\,\wedge\,\left(p\,\vee\,q\right)\right]\,\longrightarrow\,q$$ is a tautology without using truth tables.

Homework Equations

DeMorgan's Laws, etc.

The Attempt at a Solution

$$\left[\neg\,p\,\wedge\,\left(p\,\vee\,q\right)\right]\,\longrightarrow\,q$$

by. EX 3 (see EX 8)

$$\left[\neg\,p\,\wedge\,\left(p\,\vee\,q\right)\right]\,\vee\,q$$

$$\left[p\,\wedge\,\neg\,\left(p\,\vee\,q\right)\right]\,\vee\,q$$

$$\left[p\,\wedge\,\left(\neg\,p\,\wedge\,\neg\,q\right)\right]\,\vee\,q$$

$$\left[\left(p\,\wedge\,\neg\,p\right)\,\wedge\,\left(p\,\wedge\,\neg\,q\right)\right]\,\vee\,q$$

$$\left[F\,\wedge\,\left(p\,\wedge\,\neg\,q\right)\right]\,\vee\,q$$

Now what?

 

[JonF]

there is an error in your first line

a -> b is logicaly equivlent to ~a or b

 

[Architeuthis Dux]

To show that this conditional statement is a tautology, we need to show that it is always true, regardless of the truth values of the propositions p and q. We can do this by using logical equivalences and properties.

First, we can use the distributive property to rewrite the statement as:

\left[\left(p\,\wedge\,\neg\,p\right)\,\wedge\,\neg\,q\right]\,\vee\,\left[\left(p\,\wedge\,\neg\,q\right)\,\wedge\,\neg\,q\right]

Next, we can use the identity property to simplify the first part to \left[F\,\wedge\,\neg\,q\right], which is always false. This means that the entire first part of the statement is always false.

Then, we can use the distributive property again to simplify the second part to \left[p\,\wedge\,\neg\,q\right].

Finally, using the transitive property, we can rewrite the statement as \left[p\,\wedge\,\neg\,q\right]\,\vee\,q.

Using the commutative property, we can rearrange this to \left[q\,\vee\,p\right]\,\wedge\,\neg\,q.

Then, using the identity property, we can simplify this to \neg\,q.

Since \neg\,q is logically equivalent to q\,\longrightarrow\,q, we have shown that the original conditional statement is a tautology.