Mathematical Logic 12th Science Pdf

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical LogicMiscellaneous Exercise 1 [Pages 32 - 35]

Miscellaneous Exercise 1 | Q 1.1 | Page 32

Select and write the correct answer from the given alternative of the following question:

If p ∧ q is false and p ∨ q is true, then __________ is not true.

p ∨ q
p ↔ q
∼p ∨ ∼q
q ∨ ∼p

Miscellaneous Exercise 1 | Q 1.2 | Page 32

Select and write the correct answer from the given alternative of the following question:

(p ∧ q) → r is logically equivalent to ________ .

p → (q → r)
(p ∧ q) → ∼r
(∼p ∨ ∼q) → ∼r
(p ∨ q) → r

Miscellaneous Exercise 1 | Q 1.3 | Page 32

Select and write the correct answer from the given alternative of the following question:

Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .

(p ∧ q) → (p ∨ q)
∼ (p ∨ q) → (p ∧ q)
(∼p ∧ ∼q) → (∼p ∨ ∼q)
(∼p ∨ ∼q) → (∼p ∧ ∼q)

Miscellaneous Exercise 1 | Q 1.4 | Page 32

Select and write the correct answer from the given alternative of the following question:

If p ∧ q is F, p → q is F then the truth values of p and q are ________.

T, T
T, F
F, T
F, F

Miscellaneous Exercise 1 | Q 1.5 | Page 32

Select and write the correct answer from the given alternative of the following question:

The negation of inverse of ∼p → q is ________.

q ∧ p
∼p ∧ ∼q
p ∧ q
∼q → ∼p

Miscellaneous Exercise 1 | Q 1.6 | Page 32

Select and write the correct answer from the given alternative of the following question:

The negation of p ∧ (q → r) is ________.

∼p ∧ (∼q → ∼r)
p ∨ (∼q ∨ r)
∼p ∧ (∼q → ∼r)
∼p ∨ (∼q ∧ ∼r)

Miscellaneous Exercise 1 | Q 1.7 | Page 32

Select and write the correct answer from the given alternative of the following question:

If A = {1, 2, 3, 4, 5} then which of the following is not true?

∃ x ∈ A such that x + 3 = 8
∃ x ∈ A such that x + 2 < 9
∀ x ∈ A, x + 6 ≥ 9
∃ x ∈ A such that x + 6 < 10

Miscellaneous Exercise 1 | Q 2.1 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

4! = 24.

Miscellaneous Exercise 1 | Q 2.2 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

π is an irrational number.

Miscellaneous Exercise 1 | Q 2.3 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

India is a country and Himalayas is a river.

Miscellaneous Exercise 1 | Q 2.4 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

Please get me a glass of water.

Miscellaneous Exercise 1 | Q 2.5 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

cos²θ − sin²θ = cos2θ for all θ∈R.

Miscellaneous Exercise 1 | Q 2.6 | Page 33

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

If x is a whole number then x + 6 = 0.

Miscellaneous Exercise 1 | Q 3.1 | Page 33

Write the truth value of the following statement:

`sqrt5` is an irrational but `3sqrt5` is a complex number.

Miscellaneous Exercise 1 | Q 3.2 | Page 33

Write the truth value of the following statement:

∀ n ∈ N, n² + n is even number while n² – n is an odd number.

Miscellaneous Exercise 1 | Q 3.3 | Page 33

Write the truth value of the following statement:

∃ n ∈ N such that n + 5 > 10.

Miscellaneous Exercise 1 | Q 3.4 | Page 33

Write the truth value of the following statement:

The square of any even number is odd or the cube of any odd number is odd.

Miscellaneous Exercise 1 | Q 3.5 | Page 33

Write the truth value of the following statement:

In ΔABC if all sides are equal then its all angles are equal.

Miscellaneous Exercise 1 | Q 3.6 | Page 33

Write the truth value of the following statement:

∀ n ∈ N, n + 6 > 8.

Miscellaneous Exercise 1 | Q 4.1 | Page 33

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A such that x + 8 = 15

Miscellaneous Exercise 1 | Q 4.2 | Page 33

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, x + 5 < 12.

Miscellaneous Exercise 1 | Q 4.3 | Page 33

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A, such that x + 7 ≥ 11.

Miscellaneous Exercise 1 | Q 4.4 | Page 33

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, 3x ≤ 25.

Miscellaneous Exercise 1 | Q 5.1 | Page 33

Write the negation of the following:

∀ n ∈ A, n + 7 > 6.

Miscellaneous Exercise 1 | Q 5.2 | Page 33

Write the negation of the following:

∃ x ∈ A, such that x + 9 ≤ 15.

Miscellaneous Exercise 1 | Q 5.3 | Page 33

Write the negation of the following:

Some triangles are equilateral triangle.

Miscellaneous Exercise 1 | Q 6.1 | Page 33

Construct the truth table of the following:

p → (q → p)

Miscellaneous Exercise 1 | Q 6.2 | Page 33

Construct the truth table of the following:

(∼p ∨ ∼q) ↔ [∼(p ∧ q)]

Miscellaneous Exercise 1 | Q 6.3 | Page 33

Construct the truth table of the following:

∼ (∼p ∧ ∼q) ∨ q

Miscellaneous Exercise 1 | Q 6.4 | Page 33

Construct the truth table of the following:

[(p ∧ q) ∨ r] ∧ [∼r ∨ (p ∧ q)]

Miscellaneous Exercise 1 | Q 6.5 | Page 33

Construct the truth table of the following:

[(∼p ∨ q) ∧ (q → r)] → (p → r)

Miscellaneous Exercise 1 | Q 7.1 | Page 33

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

[(p → q) ∧ ∼ q] → ∼ p

Miscellaneous Exercise 1 | Q 7.2 | Page 33

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

[(p ∨ q) ∧ ∼p] ∧ ∼q

Miscellaneous Exercise 1 | Q 7.3 | Page 33

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

(p → q) ∧ (p ∧ ∼q)

Miscellaneous Exercise 1 | Q 7.4 | Page 33

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[p → (q → r)] ↔ [(p ∧ q) → r]

Miscellaneous Exercise 1 | Q 7.5 | Page 33

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[(p ∧ (p → q)] → q

Miscellaneous Exercise 1 | Q 7.6 | Page 33

Determine whether the following statement pattern is a tautology, contradiction or contingency:

(p ∧ q) ∨ (∼p ∧ q) ∨ (p ∨ ∼q) ∨ (∼p ∧ ∼q)

Miscellaneous Exercise 1 | Q 7.7 | Page 33

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[(p ∨ ∼q) ∨ (∼p ∧ q)] ∧ r

Miscellaneous Exercise 1 | Q 7.8 | Page 33

Determine whether the following statement pattern is a tautology, contradiction or contingency:

(p → q) ∨ (q → p)

Miscellaneous Exercise 1 | Q 8.1 | Page 34

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∧ q) is T

Miscellaneous Exercise 1 | Q 8.2 | Page 34

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∨ q) → q is F

Miscellaneous Exercise 1 | Q 8.3 | Page 34

Determine the truth values of p and q in the following case:

(p ∧ q) is F and (p ∧ q) → q is T

Miscellaneous Exercise 1 | Q 9.1 | Page 34

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Miscellaneous Exercise 1 | Q 9.2 | Page 34

Using truth table, prove the following logical equivalence :

(p ∧ q) → r ≡ p → (q → r)

Miscellaneous Exercise 1 | Q 10.1 | Page 34

Using rules in logic, prove the following:

p ↔ q ≡ ∼(p ∧ ∼q) ∨ ∼(q ∧ ∼p)

Miscellaneous Exercise 1 | Q 10.2 | Page 34

Using rules in logic, prove the following:

∼p ∧ q ≡ (p ∨ q) ∧ ∼p

Miscellaneous Exercise 1 | Q 10.3 | Page 34

Using rules in logic, prove the following:

∼ (p ∨ q) ∨ (∼p ∧ q) ≡ ∼p

Miscellaneous Exercise 1 | Q 11.1 | Page 34

Using the rules in logic, write the negation of the following:

(p ∨ q) ∧ (q ∨ ∼r)

Miscellaneous Exercise 1 | Q 11.2 | Page 34

Using the rules in logic, write the negation of the following:

p ∧ (q ∨ r)

Miscellaneous Exercise 1 | Q 11.3 | Page 34

Using the rules in logic, write the negation of the following:

(p → q) ∧ r

Miscellaneous Exercise 1 | Q 11.4 | Page 34

Using the rules in logic, write the negation of the following:

(∼p ∧ q) ∨ (p ∧ ∼q)

Miscellaneous Exercise 1 | Q 12.1 | Page 34

Express the following circuit in the symbolic form. Prepare the switching table:

Miscellaneous Exercise 1 | Q 12.2 | Page 34

Express the following circuit in the symbolic form. Prepare the switching table:

Miscellaneous Exercise 1 | Q 13.1 | Page 34

Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit.

Miscellaneous Exercise 1 | Q 13.2 | Page 34

Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit.

Miscellaneous Exercise 1 | Q 14.1 | Page 35

Check whether the following switching circuits are logically equivalent - Justify.

(i)

(ii)

Miscellaneous Exercise 1 | Q 14.2 | Page 35

Check whether the following switching circuits are logically equivalent - Justify.

(i)

(ii)

Miscellaneous Exercise 1 | Q 15 | Page 35

Give alternative arrangement of the switching following circuit, has minimum switches.

Miscellaneous Exercise 1 | Q 16 | Page 35

Simplify the following so that the new circuit.

Miscellaneous Exercise 1 | Q 17 | Page 35

Represent the following switching circuit in symbolic form and construct its switching table. Write your conclusion from the switching table.