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Chapter 31: Mathematical Reasoning – Exercise 31.2 Mathematical Reasoning – Exercise – 31.2 – Q.1 The negation of the statement: Banglore is the capital of Karnataka. is Banglore is not the capital of Karnataka. The negation of the statement: It rained on July 4,2005. is It did not rain on July 4, 2005. The negation of the statement: Ravish is honest. is Ravish is not honest. The negation of the statement: The earth is round. is The earth is not round. The negation of the statement: The sun is cold. is The sun is not cold. Mathematical Reasoning – Exercise – 31.2 – Q.2 The negation of the statements: All birds sing. is Not all birds sing. The negation of the statements: Some even integers are prime. is No even integers is prime. The negation of the statements: There is a complex number which is not a real number. is All complex numbers are real numbers. The negation of the statements: I will not go to school. is I will go to school. The negation of the statements: Both the diagonals of a rectangle have the same length. is There is at least one rectangle whose both diagonals do not have the same length. The negation of the statements: All policemen are thieves. is No policemen is thief. Mathematical Reasoning – Exercise – 31.2 – Q.3 (i) The number x is not a rational number. ⇒ The number x is an irrational number. ∴ The statement “The number x is not an irrational number." is a negation of the first statement. (ii) The number x is not a rational number. ⇒ The number x is an irrational number. The statement "The number x is an irrational number” is not a negation of the first statement. Mathematical Reasoning – Exercise – 31.2 – Q.4 (i) The negation of the statement: p: For every positive real number x, the number (x -1) is also positive. is ~p: There exists a positive real number x such that the number (x - 1) is not positive. (ii) The negation of the statement: q: For every real number x, either x > 1 or x < 1. is ~q: There exists a real number such that neither x > 1 or x <1. (iii) The negation of the statement: r: There are exists a number x such that 0 < x <1. is ~r: For every real number x, either x ≤ or x ≥ al. Mathematical Reasoning – Exercise – 31.2 – Q.5 The negation of the statement: a + b = b + a is true for every real number a and b. is There exist real numbers a and b for which a + b ≠ b + a. So, the given statement is not the negation of the first statement.
The negation of the statement:
Banglore is the capital of Karnataka.
is
Banglore is not the capital of Karnataka.
It rained on July 4,2005.
It did not rain on July 4, 2005.
Ravish is honest.
Ravish is not honest.
The earth is round.
The earth is not round.
The sun is cold.
The sun is not cold.
The negation of the statements:
All birds sing.
Not all birds sing.
Some even integers are prime.
No even integers is prime.
There is a complex number which is not a real number.
All complex numbers are real numbers.
I will not go to school.
I will go to school.
Both the diagonals of a rectangle have the same length.
There is at least one rectangle whose both diagonals do not have the same length.
All policemen are thieves.
No policemen is thief.
(i) The number x is not a rational number.
⇒ The number x is an irrational number.
∴ The statement “The number x is not an irrational number." is a negation of the first statement.
(ii) The number x is not a rational number.
The statement "The number x is an irrational number” is not a negation of the first statement.
(i)
p: For every positive real number x, the number (x -1) is also positive.
~p: There exists a positive real number x such that the number (x - 1) is not positive.
(ii)
q: For every real number x, either x > 1 or x < 1.
~q: There exists a real number such that neither x > 1 or x <1.
(iii)
r: There are exists a number x such that 0 < x <1.
~r: For every real number x, either x ≤ or x ≥ al.
a + b = b + a is true for every real number a and b.
There exist real numbers a and b for which a + b ≠ b + a.
So, the given statement is not the negation of the first statement.
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Chapter 31: Mathematical Reasoning –...
