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Chapters
1.2: Matrices
1.3: Differentiation
1.4: Applications of Derivatives
1.5: Integration
1.6: Definite Integration
1.7: Application of Definite Integration
1.8: Differential Equation and Applications
2.1: Commission, Brokerage and Discount
2.2: Insurance and Annuity
2.3: Linear Regression
2.4: Time Series
2.5: Index Numbers
2.6: Linear Programming
2.7: Assignment Problem and Sequencing
2.8: Probability Distributions
![SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.1 - Mathematical Logic - Shaalaa.com](/images/mathematics-and-statistics-commerce-english-12-standard-hsc_6:5f2b1b2038084cf381bfa42c826a928c.jpg "SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.1 - Mathematical Logic")
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Solutions for Chapter 1.1: Mathematical Logic
Below listed, you can find solutions for Chapter 1.1 of Maharashtra State Board SCERT Maharashtra for Mathematics and Statistics (Commerce) [English] 12 Standard HSC.
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.1 Mathematical Logic Q.1
MCQ
Choose the correct alternative :
Which of the following statement is true?
3 + 7 = 4 or 3 – 7 = 4
If Pune is in Maharashtra, then Hyderabad is in Kerala
It is false that 12 is not divisible by 3
The square of any odd integer is even
Choose the correct alternative :
Which of the following is not a statement?
2 + 2 = 4
2 is the only even prime number
Come here
Mumbai is not in Maharashtra
Choose the correct alternative:
If p is any statement, then (p ˅ ~p) is a
Contingency
Contradiction
Tautology
None of these
If p and q are two statements then (p → q) ↔ (∼ q → ∼ p) is ______.
Contradiction
Tautology
Neither (i) not (ii)
None of the these
Choose the correct alternative:
Negation of p → (p ˅ ~q) is
~p → (~p ˅ q)
p ˄ (~p ˄ q)
~p ˅ (~p ˅ ~q)
~p → (~p → q)
Choose the correct alternative :
If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is
∼ p ∨ ∼ p
∼ (p ∧ q)
∼ (p ∨ q)
p ∨ ∼ q
Choose the correct alternative:
A biconditional statement is the conjunction of two ______ statements
Negative
Compound
Connective
Conditional
Choose the correct alternative:
If p → q is an implication, then the implication ~q → ~p is called its
Converse
Contrapositive
Inverse
Alternative
The dual of the statement (p ˅ q) ˄ (r ˅ s) is ______.
(p ˄ q) ˄ (r ˄ s)
(p ˄ q) ˅ (r ˄ s)
(p ˅ q) ˅ (r ˅ s)
(p ˅ q) ˄ (r ˅ s)
The false statement in the following is ______.
p ˄ (∼ p) is contradiction
(p → q) ↔ (∼ q → ∼ p) is a contradiction
∼ (∼ p) ↔ p is a tautology
p ˅ (∼ p) ↔ p is a tautology
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.1 Mathematical Logic Q.2
State whether the following statement is True or False:
The negation of 10 + 20 = 30 is, it is false that 10 + 20 ≠ 30.
True
False
State whether the following statement is True or False :
x2 = 25 is true statement.
True
False
State whether the following statement is True or False:
p → q is equivalent to p → ~ q
True
False
State whether the following statement is True or False:
Truth value of `sqrt(3)` is not an irrational number is F
True
False
State whether the following statement is True or False:
(p ˅ q) ˄ ~ p is a contradiction
True
False
State whether the following statement is True or False:
p ↔ q is false when p and q have different truth values
True
False
State whether the following statement is True or False:
The dual of (p ˄ q) ˅ ~ q is (p ˅ q) ˄ ~ q
True
False
State whether the following statement is True or False:
Mathematical identities are true statements
True
False
State whether the following statement is True or False:
p ˅ ~ p ≡ ~ c
True
False
State whether the following statement is True or False:
The converse of inverse of ~ p → q is q → ~ p
True
False
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.1 Mathematical Logic Q.3
Fill in the following blanks
Fill in the blanks :
Conjunction of two statement p and q is symbolically written as ______.
Negation of “Some men are animal “ is ______
The truth value of negation of “London is in England” is ______
The truth value of the statement “Neither 27 is a prime number nor divisible by 4” is ______
The contrapositive of p → ~ q is ______
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.1 Mathematical Logic Q.4
Answer the following questions
Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°”
Write the following statement in symbolic form.
Milk is white if and only if the sky is not blue.
Write the following statements in symbolic form.
If Qutub – Minar is in Delhi then Taj-Mahal is in Agra
Write the following statement in symbolic form.
Even though it is not cloudy, it is still raining.
Use quantifiers to convert the given open sentence defined on N into a true statement
n2 ≥ 1
Use quantifiers to convert the given open sentence defined on N into a true statement.
3x – 4 < 9
Use quantifiers to convert the given open sentence defined on N into a true statement.
Y + 4 > 6
Examine whether the following statement pattern is a tautology, a contradiction or a contingency.
(p ∧ ~ q) → (~ p ∧ ~ q)
Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p
Write the dual of the following.
13 is prime number and India is a democratic country
Write the dual of the following
(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)
Write the converse, inverse, and contrapositive of the following statement.
"If it snows, then they do not drive the car"
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.1 Mathematical Logic Q.5
Answer the following questions
Examine whether the statement pattern
[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.
Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).
Without using truth table show that
(p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ ( ~ p ∧ q)
With proper justification, state the negation of the following.
(p ↔ q) v (~ q → ~ r)
Prepare truth table for (p ˄ q) ˅ ~ r
(p ∧ q) ∨ ~ r
If p ↔ q and p → q both are true, then find truth values of the following with the help of activity
p ˅ q
|
p ↔ q and p → q both are true if p and q has truth value `square`, `square` or `square`, `square`. p ˅ q i. If both p and q are true, then p ˅ q = `square` ˅ `square` = `square` ii. If both p and q are false, then p ˅ q = `square` ˅ `square` = `square` |
If p ↔ q and p → q both are true, then find truth values of the following with the help of activity
p ˄ q
|
p ↔ q and p → q both are true if p and q has truth value `square`, `square` or `square`, `square` p ˄ q i. If both p and q are true, then p ˄ q = `square` ˄ `square` = `square` ii. If both p and q are false, then p ˄ q = `square` ˄ `square` = `square` |
Given following statements
p: 9 × 5 = 45
q: Pune is in Maharashtra
r: 3 is the smallest prime number
Write truth values by activity
|
i) (p ˄ q) ˄ r = `(square` ˄ `square)` ˄ `square` = `square` ˄ `square` = `square` ii) ~ ( p ˄ r ) = `~(square` ˄ `square)` = `~ square` = `square` iii) p → q = `square → square` = `square` |
Complete the truth table.
| p | q | r | q → r | r → p | (q → r) ˅ (r → p) |
| T | T | T | T | `square` | T |
| T | T | F | F | `square` | `square` |
| T | F | T | T | `square` | T |
| T | F | F | T | `square` | `square` |
| F | T | T | `square` | F | T |
| F | T | F | `square` | T | `square` |
| F | F | T | `square` | F | T |
| F | F | F | `square` | T | `square` |
The given statement pattern is a `square`
Solutions for 1.1: Mathematical Logic
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.1 - Mathematical Logic
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Concepts covered in Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.1 Mathematical Logic are Truth Value of Statement, Quantifier and Quantified Statements in Logic, Statement Patterns and Logical Equivalence, Algebra of Statements, Venn Diagrams, Logical Connective, Simple and Compound Statements.
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