Discrete Mathematics

The statement $(\sim P \leftrightarrow Q) \land \sim Q$ is true, when

(i) $P$ : True, $Q$ : False
(ii) $P$ : True, $Q$ : True
(iii) $P$ : False, $Q$ : True
(iv) $P$ : False, $Q$ : False

(a) Prove the following:

(i) There is a real number $x$ such that $x > 1$ and $2^x > x^{10}$.
(ii) There is an integer $n$ such that $2n^2 - 5n + 2$ is prime.

Disprove that for all real numbers $a$ and $b$, if $a < b$, then $a^2 < b^2$.

(a) Show that $p \lor (q \land r)$ and $(p \lor q) \land (p \lor r)$ are logically equivalent. This is the distributive law of disjunction over conjunction.

Write short notes on the following:

(i) Forward proof
(ii) Disjunctive and conjunctive normal form
(iii) Fundamental theorem of arithmetic

This question covers topics from different modules:

• (i) Forward proof: This falls under Module 3: Propositional Logic & Proof Techniques.
• (ii) Disjunctive and conjunctive normal form: This also falls under Module 3: Propositional Logic & Proof Techniques, as these are concepts in propositional logic.
• (iii) Fundamental theorem of arithmetic: This is a concept from number theory, which is typically covered in Module 2: Mathematical Induction & Counting Techniques (as number theory is often grouped with induction and counting in discrete mathematics courses).

Check the validity of the following argument all integers are rational numbers. Some integers are

powers of 5. There fore, some rational numbers are powers of 5

The statement p → q is logically equivalent to
(i) p ∨ q
(ii) p ∨ ~q
(iii) ~p ∨ q
(iv) ~p → q

The contrapositive of the conditional statement $p \rightarrow q$ is
(i) $q \rightarrow p$
(ii) $\sim p \rightarrow \sim q$
(iii) $p \rightarrow q$
(iv) $\sim q \rightarrow \sim p$

Using truth table, show that-
(i) ((p → (q → r)) → ((p → q) → (p → r))) is a tautology;
(ii) ¬(q → r) ∧ r ∧ (p → q) is a contradiction.

Obtain the principal disjunctive normal form (PDNF) and principal conjunctive normal form (PCNF) of the statement
(p → (q ∧ r)) ∧ (¬p → (¬q ∧ ¬r)).