Tamil Nadu Board Solutions for Chapter: Discrete Mathematics, Exercise 2: Exercise 2

Which one of the following sentences is a proposition?

(i)$4+7=12$

(ii) What are you doing?

(iii)$3^n\le 81,n\in \mathrm{\mathbb{N}}$

(iv) Peacock is our national bird.

(v) How tall this mountain is!

Prove that $q\to p\equiv \neg p\to \neg q$

Show that $p\to q$ and $q\to p$ are not equivalent

Show that $\neg \left(p\leftrightarrow q\right)\equiv p\leftrightarrow \neg q$

Check whether the statement $p\to \left(q\to p\right)$ is a tautology or a contradiction without using the truth table.

Using  truth table check whether the statements $\neg (p\vee q)\vee (\neg p\wedge q)$ and $\neg p$ are logically equivalent.

Prove $p\to (q\to r)\equiv (p\wedge q)\to r$ without using truth table.

Prove that $p\to \left(\neg q\vee r\right)\equiv \left(\neg p\vee \neg q\right)\vee r$ using truth table.