Advertisements
Advertisements
Question
Show that the following statement is true by the method of contrapositive.
p: If x is an integer and x2 is even, then x is also even.
Advertisements
Solution
p: If x is an integer and x2 is even, then x is also even.
Let q: x is an integer and x2 is even.
r: x is even.
To prove that p is true by contrapositive method, we assume that r is false, and prove that q is also false.
Let x is not even.
To prove that q is false, it has to be proved that x is not an integer or x2 is not even.
x is not even implies that x2 is also not even.
Therefore, statement q is false.
Thus, the given statement p is true.
RELATED QUESTIONS
Find out the sentence are statement and are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Find out the sentence are statement and are not. Justify your answer.
The cat pussy is black.
Find out the sentence are statement and are not. Justify your answer.
All triangles have three sides.
Find out the sentence are statement and are not. Justify your answer.
There are 35 days in a month.
Find out the sentence are statement and are not. Justify your answer.
The product of (−1) and 8 is 8.
Write the negation of the statement:
Ravish is honest.
Check whether the following pair of statements are negation of each other. Give reasons for your answer.
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.
Find the component statement of the compound statement:
The sky is blue and the grass is green.
Find the component statement of the compound statement:
25 is a multiple of 5 and 8.
For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.
To entry a country, you need a passport or a voter registration card.
Write the negation of statement:
There exists x ϵ N, x + 3 = 10
Check the validity of the statement:
p : 100 is a multiple of 4 and 5.
Which of the following statements are compound statements
“9 is neither an even number nor a prime number”
Write the truth value of the following statements.
(i) 9 is an even integer or 9 + 1 is even.
(ii) 2 + 4 = 6 or 2 + 4 = 7
(iii) Delhi is the capital of India and Islamabad is the capital of Pakistan.
(iv) Every rectangle is a square and every square is a rectangle
(v) The sun is a star or sun is a planet.
Rewrite the following statements in the form of conditional statements:
When you sing, my ears hurt.
Rewrite the following statements in the form of conditional statements:
A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan.
Translate the following statements in symbolic form:
If x = 7 and y = 4” then x + y = 11.
Translate the following biconditional into symbolic form:
“ABC is an equilateral triangle if and only if its each interior angle is 60°”
Identify the quantifiers and write the negation of the following statements:
There exists a number which is equal to its square.
Identify the quantifiers and write the negation of the following statements:
For all even integers x, x2 is also even.
Check the validity of the statements:
s: 60 is a multiple of 3 or 5.
Which of the following sentences are statements? Justify
0 is a complex number.
Which of the following sentences are statements? Justify
15 + 8 > 23
Find the component statements of the following compound statements.
Chennai is in India and is the capital of Tamil Nadu.
Find the component statements of the following compound statements.
Chandigarh is the capital of Haryana and U.P.
Find the component statements of the following compound statements.
`sqrt(7)` is a rational number or an irrational number.
Find the component statements of the following compound statements.
Two lines in a plane either intersect at one point or they are parallel.
Find the component statements of the following compound statements.
A rectangle is a quadrilateral or a 5-sided polygon.
Translate the following statements into symbolic form
Rahul passed in Hindi and English.
Translate the following statements into symbolic form
Either x = 2 or x = 3 is a root of 3x 2 – x – 10 = 0
Write down the negation of following compound statements
All rational numbers are real and complex.
Write down the negation of following compound statements
x = 2 and x = 3 are roots of the Quadratic equation x2 – 5x + 6 = 0.
Form the biconditional statement p ↔ q, where
p: A natural number n is odd.
q: Natural number n is not divisible by 2.
Identify the Quantifiers in the following statements.
For all real numbers x and y, xy = yx.
Identify the Quantifiers in the following statements.
There exists a statement in above statements which is not true.
Which of the following is a statement?
The negation of the statement “Rajesh or Rajni lived in Bangalore” is ______.
Which of the following is not a negation of “A natural number is greater than zero”?
