Mathematical Logic Exercise 1.6 [Page 16] Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1

EXERCISE 1.6 [PAGE 16]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.6 [Page 16]

EXERCISE 1.6 Q 1.1 | PAGE 16

Prepare truth tables for the following statement pattern.

\(p \rightarrow (\sim p \lor q)\)

SOLUTION

\(p \rightarrow (\sim p \lor q)\)

\(p\)	\(q\)	\(\sim p\)	\(\sim p \lor q\)	\(p \rightarrow (\sim p \lor q)\)
T	T	F	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

EXERCISE 1.6 Q 1.2 | PAGE 16

Prepare truth tables for the following statement pattern.

\((\sim p \lor q) \land (\sim p \lor \sim q)\)

SOLUTION

\((\sim p \lor q) \land (\sim p \lor \sim q)\)

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim p \lor q\)	\(\sim p \lor \sim q\)	\((\sim p \lor q) \land (\sim p \lor \sim q)\)
T	T	F	F	T	F	F
T	F	F	T	F	T	F
F	T	T	F	T	T	T
F	F	T	T	T	T	T

EXERCISE 1.6 Q 1.3 | PAGE 16

Prepare truth tables for the following statement pattern.

\((p \land r) \rightarrow (p \lor \sim q)\)

SOLUTION

\((p \land r) \rightarrow (p \lor \sim q)\)

\(p\)	\(q\)	\(r\)	\(\sim q\)	\(p \land r\)	\(p \lor \sim q\)	\((p \land r) \rightarrow (p \lor \sim q)\)
T	T	T	F	T	T	T
T	T	F	F	F	T	T
T	F	T	T	T	T	T
T	F	F	T	F	T	T
F	T	T	F	F	F	T
F	T	F	F	F	F	T
F	F	T	T	F	T	T
F	F	F	T	F	T	T

EXERCISE 1.6 Q 1.4 | PAGE 16

Prepare truth tables for the following statement pattern.

\((p \land q) \lor \sim r\)

SOLUTION

\((p \land q) \lor \sim r\)

\(p\)	\(q\)	\(r\)	\(\sim r\)	\(p \land q\)	\((p \land q) \lor \sim r\)
T	T	T	F	T	T
T	T	F	T	T	T
T	F	T	F	F	F
T	F	F	T	F	T
F	T	T	F	F	F
F	T	F	T	F	T
F	F	T	F	F	F
F	F	F	T	F	T

EXERCISE 1.6 Q 2.1 | PAGE 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

\(q \lor [\sim (p \land q)]\)

SOLUTION

\(p\)	\(q\)	\(p \land q\)	\(\sim (p \land q)\)	\(q \lor [\sim (p \land q)]\)
T	T	T	F	T
T	F	F	T	T
F	T	F	T	T
F	F	F	T	T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6 Q 2.2 | PAGE 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

\((\sim q \land p) \land (p \land \sim p)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\((\sim q \land p)\)	\((p \land \sim p)\)	\((\sim q \land p) \land (p \land \sim p)\)
T	T	F	F	F	F	F
T	F	F	T	T	F	F
F	T	T	F	F	F	F
F	F	T	T	F	F	F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6 Q 2.3 | PAGE 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

\((p \land \sim q) \rightarrow (\sim p \land \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \land \sim q\)	\(\sim p \land \sim q\)	\((p \land \sim q) \rightarrow (\sim p \land \sim q)\)
T	T	F	F	F	F	T
T	F	F	T	T	F	F
F	T	T	F	F	F	T
F	F	T	T	F	T	T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6 Q 2.4 | PAGE 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

\(\sim p \rightarrow (p \rightarrow \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \rightarrow \sim q\)	\(\sim p \rightarrow (p \rightarrow \sim q)\)
T	T	F	F	F	T
T	F	F	T	T	T
F	T	T	F	T	T
F	F	T	T	T	T

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6 Q 3.1 | PAGE 16

Prove that the following statement pattern is a tautology.

\((p \land q) \rightarrow q\)

SOLUTION

\(p\)	\(q\)	\(p \land q\)	\((p \land q) \rightarrow q\)
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6 Q 3.2 | PAGE 16

Prove that the following statement pattern is a tautology.

\((p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \rightarrow q\)	\(\sim q \rightarrow \sim p\)	\((p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)\)
T	T	F	F	T	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	T
F	F	T	T	T	T	T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6 Q 3.3 | PAGE 16

Prove that the following statement pattern is a tautology.

\((\sim p \land \sim q ) \rightarrow (p \rightarrow q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim p \land \sim q\)	\(p \rightarrow q\)	\((\sim p \land \sim q) \rightarrow (p \rightarrow q)\)
T	T	F	F	F	T	T
T	F	F	T	F	F	T
F	T	T	F	F	T	T
F	F	T	T	T	T	T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6 Q 3.4 | PAGE 16

Prove that the following statement pattern is a tautology.

\((\sim p \lor \sim q) \leftrightarrow \sim (p \land q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim p \lor \sim q\)	\(p \land q\)	\(\sim (p \land q)\)	\((\sim p \lor \sim q) \leftrightarrow \sim(p \land q)\)
T	T	F	F	F	T	F	T
T	F	F	T	T	F	T	T
F	T	T	F	T	F	T	T
F	F	T	T	T	F	T	T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6 Q 4.1 | PAGE 16

Prove that the following statement pattern is a contradiction.

\((p \lor q) \land (\sim p \land \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \lor q\)	\(\sim p \land \sim q\)	\((p \lor q) \land (\sim p \land \sim q)\)
T	T	F	F	T	F	F
T	F	F	T	T	F	F
F	T	T	F	T	F	F
F	F	T	T	F	T	F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6 Q 4.2 | PAGE 16

Prove that the following statement pattern is a contradiction.

\((p \land q) \land \sim p\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(p \land q\)	\((p \land q) \land \sim p\)
T	T	F	T	F
T	F	F	F	F
F	T	T	F	F
F	F	T	F	F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6 Q 4.3 | PAGE 16

Prove that the following statement pattern is a contradiction.

\((p \land q) \land (\sim p \lor \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \land q\)	\(\sim p \lor \sim q\)	\((p \land q) \land (\sim p \lor \sim q)\)
T	T	F	F	T	F	F
T	F	F	T	F	T	F
F	T	T	F	F	T	F
F	F	T	T	F	T	F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6 Q 4.4 | PAGE 16

Prove that the following statement pattern is a contradiction.

\((p \rightarrow q) \land (p \land \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim q\)	\(p \rightarrow q\)	\(p \land \sim q\)	\((p \rightarrow q) \land (p \land \sim q)\)
T	T	F	T	F	F
T	F	T	F	T	F
F	T	F	T	F	F
F	F	T	T	F	F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6 Q 5.1 | PAGE 16

Show that the following statement pattern is contingency.

\((p \land \sim q) \rightarrow (\sim p \land \sim q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \land \sim q\)	\(\sim p \land \sim q\)	\((p \land \sim q) \rightarrow (\sim p \land \sim q)\)
T	T	F	F	F	F	T
T	F	F	T	T	F	F
F	T	T	F	F	F	T
F	F	T	T	F	T	T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6 Q 5.2 | PAGE 16

Show that the following statement pattern is contingency.

\((p \rightarrow q) \leftrightarrow (\sim p \lor q)\)

SOLUTION

\(p\)	\(q\)	\(\sim p\)	\(p \rightarrow q\)	\(\sim p \lor q\)	\((p \rightarrow q) \leftrightarrow (\sim p \lor q)\)
T	T	F	T	T	T
T	F	F	F	F	T
F	T	T	T	T	T
F	F	T	T	T	T

All the truth values in the last column are T. Hence, it is a tautology. Not contingency.

EXERCISE 1.6 Q 5.3 | PAGE 16

Show that the following statement pattern is contingency.

\(p \land [(p \rightarrow \sim q) \rightarrow q]\)

SOLUTION

\(p\)	\(q\)	\(\sim q\)	\(p \rightarrow \sim q\)	\((p \rightarrow \sim q) \rightarrow q\)	\(p \land [(p \rightarrow \sim q) \rightarrow q]\)
T	T	F	F	T	T
T	F	T	T	F	F
F	T	F	T	T	F
F	F	T	T	F	F

Truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6 Q 5.4 | PAGE 16

Show that the following statement pattern is contingency.

\((p \rightarrow q) \land (p \rightarrow r)\)

SOLUTION

\(p\)	\(q\)	\(r\)	\(p \rightarrow q\)	\(p \rightarrow r\)	\((p \rightarrow q) \land (p \rightarrow r)\)
T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	F
T	F	F	F	F	F
F	T	T	T	T	T
F	T	F	T	T	T
F	F	T	T	T	T
F	F	F	T	T	T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6 Q 6.1 | PAGE 16

Using the truth table, verify

\(p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(r\)	4 \(q \land r\)	5 \(p \lor (q \land r)\)	6 \(p \lor q\)	7 \(p \lor r\)	8 \((p \lor q) \land (p \lor r)\)
T	T	T	T	T	T	T	T
T	T	F	F	T	T	T	T
T	F	T	F	T	T	T	T
T	F	F	F	T	T	T	T
F	T	T	T	T	T	T	T
F	T	F	F	F	T	F	F
F	F	T	F	F	F	T	F
F	F	F	F	F	F	F	F

The entries in columns 5 and 8 are identical.

\(\therefore p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\)

EXERCISE 1.6 Q 6.2 | PAGE 16

Using the truth table, verify

\(p \rightarrow (p \rightarrow q) \equiv \sim q \rightarrow (p \rightarrow q)\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(\sim q\)	4 \(p \rightarrow q\)	5 \(p \rightarrow (p \rightarrow q)\)	6 \(\sim q \rightarrow (p \rightarrow q)\)
T	T	F	T	T	T
T	F	T	F	F	F
F	T	F	T	T	T
F	F	T	T	T	T

In the above truth table, entries in columns 5 and 6 are identical.

\(\therefore p \rightarrow (p \rightarrow q) \equiv \sim q \rightarrow (p \rightarrow q)\)

EXERCISE 1.6 Q 6.3 | PAGE 16

Using the truth table, verify

\(\sim(p \rightarrow \sim q) \equiv p \land \sim (\sim q) \equiv p \land q\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(\sim q\)	4 \(p \rightarrow \sim q\)	5 \(\sim(p \rightarrow \sim q)\)	6 \(\sim (\sim q)\)	7 \(p \land \sim (\sim q)\)	8 \(p \land q\)
T	T	F	F	T	T	T	T
T	F	T	T	F	F	F	F
F	T	F	T	F	T	F	F
F	F	T	T	F	F	F	F

In the above table, entries in columns 5, 7, and 8 are identical.

\(\therefore \sim(p \rightarrow \sim q) \equiv p \land \sim (\sim q) \equiv p \land q\)

EXERCISE 1.6 Q 6.4 | PAGE 16

Using the truth table, verify

\(\sim(p \lor q) \lor (\sim p \land q) \equiv \sim p\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(\sim p\)	4 \((p \lor q)\)	5 \(\sim(p \lor q)\)	6 \(\sim p \land q\)	7 \(\sim(p \lor q) \lor (\sim p \land q)\)
T	T	F	T	F	F	F
T	F	F	T	F	F	F
F	T	T	T	F	T	T
F	F	T	F	T	F	T

In the above truth table, the entries in columns 3 and 7 are identical.

\(\therefore \sim(p \lor q) \lor (\sim p \land q) \equiv \sim p\)

EXERCISE 1.6 Q 7.1 | PAGE 16

Using the truth table, verify

\(p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(r\)	4 \(q \land r\)	5 \(p \lor (q \land r)\)	6 \(p \lor q\)	7 \(p \lor r\)	8 \((p \lor q) \land (p \lor r)\)
T	T	T	T	T	T	T	T
T	T	F	F	T	T	T	T
T	F	T	F	T	T	T	T
T	F	F	F	T	T	T	T
F	T	T	T	T	T	T	T
F	T	F	F	F	T	F	F
F	F	T	F	F	F	T	F
F	F	F	F	F	F	F	F

The entries in columns 5 and 8 are identical.

\(\therefore p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\)

EXERCISE 1.6 Q 7.2 | PAGE 16

Prove that the following pair of statement pattern is equivalent.

\(p \leftrightarrow q\) and \((p \rightarrow q) \land (q \rightarrow p)\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(p \leftrightarrow q\)	4 \(p \rightarrow q\)	5 \(q \rightarrow p\)	6 \((p \rightarrow q) \land (q \rightarrow p)\)
T	T	T	T	T	T
T	F	F	F	T	F
F	T	F	T	F	F
F	F	T	T	T	T

In the above table, entries in columns 3 and 6 are identical.

\(\therefore\) Statement \(p \leftrightarrow q\) and \((p \rightarrow q) \land (q \rightarrow p)\) are equivalent.

EXERCISE 1.6 Q 7.3 | PAGE 16

Prove that the following pair of statement pattern is equivalent.

\(p \rightarrow q\) and \(\sim q \rightarrow \sim p\) and \(\sim p \lor q\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(\sim p\)	4 \(\sim q\)	5 \(p \rightarrow q\)	6 \(\sim q \rightarrow \sim p\)	7 \(\sim p \lor q\)
T	T	F	F	T	T	T
T	F	F	T	F	F	F
F	T	T	F	T	T	T
F	F	T	T	T	T	T

In the above table, entries in columns 5, 6 and 7 are identical.

\(\therefore\) Statement \(p \rightarrow q\) and \(\sim q \rightarrow \sim p\) and \(\sim p \lor q\) are equivalent.

EXERCISE 1.6 Q 7.4 | PAGE 16

Prove that the following pair of statement pattern is equivalent.

\(\sim(p \land q)\) and \(\sim p \lor \sim q\)

SOLUTION

1 \(p\)	2 \(q\)	3 \(\sim p\)	4 \(\sim q\)	5 \(p \land q\)	6 \(\sim(p \land q)\)	7 \(\sim p \lor \sim q\)
T	T	F	F	T	F	F
T	F	F	T	F	T	T
F	T	T	F	F	T	T
F	F	T	T	F	T	T

In the above table, entries in columns 6 and 7 are identical.

\(\therefore\) Statement \(\sim(p \land q)\) and \(\sim p \lor \sim q\) are equivalent.