Exercise 2.1 | Advanced Mathematics | Class 9

Propositional Logic|Advanced Mathematics|Class 9

Exercise 2.1 | Advanced Mathematics | Class 9

1. Fill up the tables of truth values -
(A)
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \vee{q} \) \(\ p \wedge{q} \) \(\ \sim({p \vee{q}}) \) \(\ \sim{p \wedge{q}} \) \(\ \sim{p} \vee \sim{q} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ F \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)

(B)
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \wedge{\sim{q}} \) \(\ p \to{q} \) \(\ \sim{(p \to{q})} \) \(\ \sim{p} \to \sim{q} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ F \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)

(C)
\(\ p \) \(\ q \) \(\ p \to{q} \) \(\ p \wedge{(p \to{q})} \) \(\ p \wedge{(p \to{q})} \to{q} \)
\(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)



2. Prove all the algebraic properties of statements with the help of truth tables.
Solution: Click Here


3. If \(\ a \), \(\ b \) and \(\ c \) are any three statements, then prove that-
(a) \(\ (a \wedge b) \to{(a \vee{b})} \)
(b) \(\ [(a \to{b}) \wedge{(b \to{c})}] \to{(a \to{c} )} \)
are two statements both of which are formula (or tentologies).
Solution:
(a)
\(\ a \) \(\ b \) \(\ a \wedge{b} \) \(\ a \vee{b} \) \(\ (a \wedge{b}) \to{(a \vee{b})} \)
\(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ F \) \(\ F \) \(\ F \) \(\ T \)

(b)
\(\ a \) \(\ b \) \(\ c \) \(\ a \to{b} \) \(\ b \to{c} \) \(\ a \to{c} \) \(\ (a \to{b}) \wedge{(b \to{c})} \) \(\ [(a \to{b}) \wedge{(b \to{c})}] \to{(a \to{c})} \)
\(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
\(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ T \)
\(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
\(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
\(\ F \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)



4. Prove that
(a) \(\ \sim{(\sim{p}\wedge{\sim{q}}) \equiv p \vee q} \)
Solution:
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \vee{q} \) \(\ \sim{p} \wedge{\sim{q}} \) \(\ \sim{(\sim{p} \wedge{\sim{q}})} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \)

(b) \(\ \sim{(\sim{p}\to{\sim{q}}) \equiv \sim{p} \vee q} \)
Solution:
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ \sim{p}\to{\sim{q}} \) \(\ \sim{(\sim{p}\to{\sim{q}})} \) \(\ \sim{p} \wedge{q} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ F \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ F \) \(\ F \)



5. Test if correct or incorrect-
(a) \(\ p \to{q} \equiv{\sim{p \to \sim{q}}} \)
Solution:
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \to{q} \) \(\ \sim{p} \to{\sim{q}} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ T \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ T \)
It is not correct.

(b) \(\ \sim{(p \to{q})} \equiv{p \to{\sim{q}}} \)
Solution:
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \to{q} \) \(\ \sim{(p \to{q})} \) \(\ p \to{\sim{q}} \)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ F \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ F \) \(\ T \)
It is not correct.

(c) \(\ \sim{(p \to{q})} \equiv{(p \to{\sim{q}}) \vee{(q \wedge{\sim{p}})}} \)
Solution:
\(\ p \) \(\ q \) \(\ \sim{p} \) \(\ \sim{q} \) \(\ p \to{q} \) \(\ \sim{(p \to{q})} \) \(\ p \to{\sim{q}} \) \(\ q \wedge{\sim{p}} \) \(\ (p \to{\sim{q}}) \vee{(q \wedge{\sim{p}})}\)
\(\ T \) \(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ F \) \(\ F \) \(\ F \)
\(\ T \) \(\ F \) \(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \)
\(\ F \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \) \(\ T \) \(\ T \)
\(\ F \) \(\ F \) \(\ T \) \(\ T \) \(\ T \) \(\ F \) \(\ T \) \(\ F \) \(\ T \)
It is not correct.
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