### Mathematical Logic - Maharashtra Board 12th Maths Solutions Chapter 1 Ex 1.2

### Question 1.Construct the truth table for each of the following statement patterns:

**(i) [(p → q) ∧ q] → p**

**Solution :**

Here are two statements and three connectives.

∴ there are 2 × 2 = 4 rows and 2 + 3 = 5 columns in the truth table.

**(ii) (p ∧ ~q) ↔ (p → q)**

**Solution:**

**(iii) (p ∧ q) ↔ (q ∨ r)**

**Solution:**

**(iv) p → [~(q ∧ r)]**

**Solution:**

**(v) ~p ∧ [(p ∨ ~q ) ∧ q]**

**Solution:**

**(vi) (~p → ~q) ∧ (~q → ~p)**

**Solution:**

**(vii) (q → p) ∨ (~p ↔ q)**

**Solution:**

**(viii) [p → (q → r)] ↔ [(p ∧ q) → r]**

**Solution:**

**(ix) p → [~(q ∧ r)]**

**Solution:**

**(x) (p ∨ ~q) → (r ∧ p)**

**Solution:**

### Question 2.Using truth tables prove the following logical equivalences.

**(i) ~p ∧ q ≡ (p ∨ q) ∧ ~p**

**Solution:**

The entries in the columns 4 and 6 are identical.

∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p.

**(ii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p**

**Solution:**

The entries in the columns 3 and 7 are identical.

∴ ~(p ∨ q) ∧ (~p ∧ q) = ~p.

**(iii) p ↔ q ≡ ~[(p ∨ q) ∧ ~(p ∧ q)]**

**Solution:**

The entries in the columns 3 and 8 are identical.

∴ p ↔ q ≡ ~[(p ∨ q) ∧ ~(p ∧ q)].

**(iv) p → (q → p) ≡ ~p → (p → q)**

**Solution:**

The entries in the columns 4 and 7 are identical.

∴ p → (q → p) ≡ ~p → (p → q).

**(v) (p ∨ q ) → r ≡ (p → r) ∧ (q → r)**

**Solution:**

The entries in the columns 5 and 8 are identical.

∴ (p ∨ q ) → r ≡ (p → r) ∧ (q → r).

(vi) p → (q ∧ r) ≡ (p → q) ∧ (p → r)

Solution:

The entries in the columns 5 and 8 are identical.

∴ p → (q ∧ r) ≡ (p → q) ∧ (p → r).

**(vii) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)**

**Solution:**

The entries in the columns 5 and 8 are identical.

∴ p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).

**(viii) [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r**

**Solution:**

The entries in the columns 3 and 7 are identical.

∴ [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r.

**(ix) ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p)**

**Solution:**

The entries in the columns 6 and 9 are identical.

∴ ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p).

### Question 3.Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

**(i) (p ∧ q) → (q ∨ p)**

**Solution:**

All the entries in the last column of the above truth table are T.

∴ (p ∧ q) → (q ∨ p) is a tautology.

**(ii) (p → q) ↔ (~p ∨ q)**

**Solution:**

All the entries in the last column of the above truth table are T.

∴ (p → q) ↔ (~p ∨ q) p is a tautology.

**(iii) [~(~p ∧ ~q)] ∨ q**

**Solution:**

The entries in the last column of the above truth table are neither all T nor all F.

∴ [~(~p ∧ ~q)] ∨ q is a contingency.

**(iv) [(p → q) ∧ q)] → p**

**Solution:**

The entries in the last column of the above truth table are neither all T nor all F.

∴ [(p → q) ∧ q)] → p is a contingency

**(v) [(p → q) ∧ ~q] → ~p**

**Solution:**

All the entries in the last column of the above truth table are T.

∴ [(p → q) ∧ ~q] → ~p is a tautology.

**(vi) (p ↔ q) ∧ (p → ~q)**

**Solution:**

The entries in the last column of the above truth table are neither all T nor all F.

∴ (p ↔ q) ∧ (p → ~q) is a contingency.

**(vii) ~(~q ∧ p) ∧ q**

**Solution:**

The entries in the last column of the above truth table are neither all T nor all F.

∴ ~(~q ∧ p) ∧ q is a contingency.

**(viii) (p ∧ ~q) ↔ (p → q)**

**Solution:**

All the entries in the last column of the above truth table are F.

∴ (p ∧ ~q) ↔ (p → q) is a contradiction.

**(ix) (~p → q) ∧ (p ∧ r)**

**Solution:**

The entries in the last column of the above truth table are neither all T nor all F.

∴ (~p → q) ∧ (p ∧ r) is a contingency.

**(x) [p → (~q ∨ r)] ↔ ~[p → (q → r)]**

**Solution:**

All the entries in the last column of the above truth table are F.

∴ [p → (~q ∨ r)] ↔ ~[p → (q → r)] is a contradiction

### 1.2 STATEMENT PATTERN, LOGICAL EQUIVALENCE, TAUTOLOGY, CONTRADICTION, CONTINGENCY.

**1.2.1 Statement Pattern :**

Letters used to denote statements are called statement letters. Proper combination of statement letters and connectives is called a statement pattern. Statement pattern is also called as a proposition. p → q, p ∧ q, ~ p ∨ q are statement patterns. p and q are their prime components.

A table which shows the possible truth values of a statement pattern obtained by considering all possible combinations of truth values of its prime components is called the truth table of the statement pattern.

**1.2.2. Logical Equivalence :**

Two statement patterns are said to be equivalent if their truth tables are identical. If statement patterns A and B are equivalent, we write it as A ≡ B.

**1.2.3 Tautology, Contradiction and Contingency :**

Tautology : A statement pattern whose truth value is true for all possible combinations of truth values of its prime components is called a tautology. We denote tautology by t. Statement pattern p ∨ ~ p is a tautology.

**Contradiction :**A statement pattern whose truth value is false for all possible combinations of truth values of its prime components is called a contradiction. We denote contradiction by c. Statement pattern p ∧ ~ p is a contradiction.

**Contingency :**A statement pattern which is neither a tautology nor a contradiction is called a contingency. p ∧ q is a contingency.

### 1.2 STATEMENT PATTERN, LOGICAL EQUIVALENCE, TAUTOLOGY, CONTRADICTION, CONTINGENCY.

- mathematical logic class 1.2 exercise 1.6 solutions
- mathematical logic exercise 1.2
- mathematical logic exercise 1.3
- mathematical logic exercise 1.2 solutions
- mathematical logic exercise 1.1
- mathematical logic exercise 1.5
- mathematical logic class 12 exercise 1.2
- mathematical logic class 12 exercise 1.2 solutions
- mathematical logic exercise 1.2 new syllabus
- mathematical logic class 1.2 exercise 1.3 solutions
- mathematical logic class 12 exercise 1.3 solutions Mathematical Logic - Maharashtra Board 12th Maths Solutions Chapter 1 Ex 1.2