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Are the following pairs of statements negations of each other?
The number x is not a rational number.
The number x is not an irrational number.
Concept: undefined >> undefined
Are the following pairs of statements negations of each other?
The number x is a rational number.
The number x is an irrational number.\
Concept: undefined >> undefined
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Find the component statements of the following compound statements and check whether they are true or false.
Number 3 is prime or it is odd.
Concept: undefined >> undefined
Find the component statements of the following compound statements and check whether they are true or false.
All integers are positive or negative.
Concept: undefined >> undefined
Find the component statements of the following compound statements and check whether they are true or false.
100 is divisible by 3, 11 and 5.
Concept: undefined >> undefined
Write the negation of the following statements:
p: For every positive real number x, the number x – 1 is also positive.
Concept: undefined >> undefined
Write the negation of the following statements:
q: All cats scratch.
Concept: undefined >> undefined
Write the negation of the following statements:
r: For every real number x, either x > 1 or x < 1.
Concept: undefined >> undefined
Write the negation of the following statements:
s: There exists a number x such that 0 < x < 1.
Concept: undefined >> undefined
Which of the following are examples of empty set?
Set of all even natural numbers divisible by 5
Concept: undefined >> undefined
Which of the following are examples of empty set?
Set of all even prime numbers
Concept: undefined >> undefined
Which of the following are examples of empty set?
{x : x2 −2 = 0 and x is rational}
Concept: undefined >> undefined
Which of the following are examples of empty set?
{x : x is a natural number, x < 8 and simultaneously x > 12};
Concept: undefined >> undefined
Which of the following are examples of empty set?
{x : x is a point common to any two parallel lines}.
Concept: undefined >> undefined
If P (n) is the statement "n(n + 1) is even", then what is P(3)?
Concept: undefined >> undefined
If P (n) is the statement "n3 + n is divisible by 3", prove that P (3) is true but P (4) is not true.
Concept: undefined >> undefined
If P (n) is the statement "2n ≥ 3n" and if P (r) is true, prove that P (r + 1) is true.
Concept: undefined >> undefined
If P (n) is the statement "n2 + n is even", and if P (r) is true, then P (r + 1) is true.
Concept: undefined >> undefined
Given an example of a statement P (n) such that it is true for all n ∈ N.
Concept: undefined >> undefined
If P (n) is the statement "n2 − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.
Concept: undefined >> undefined
