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 real numbers are complex numbers.
Write the negation of the statement:
The sun is cold.
All birds sing.
There is a complex number which is not a real number.
I will not go to school.
Are the pair of statement are negation of each other:
The number x is not a rational number.
The number x is not an irrational number.
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:
The earth is round or the sun is cold.
Determine whether the compound statement are true or false:
Delhi is in England and 2 + 2 =5.
Write the negation of statement:
For every x ϵ N, x + 3 < 10
Write of the statement in the form "if p, then q".
It is necessary to have a passport to log on to the server.
statement are true and false? In each case give a valid reason for saying so
q : The centre of a circle bisects each chord of the circle.
statement are true and false? In each case give a valid reason for saying so
t : \[\sqrt{11}\] is a rational number.
Which of the following statements are compound statements
“2 is both an even number and a prime number”
Translate the following statements in symbolic form
Tigers are found in Gir forest or Rajaji national park.
Write the negation of the following statements:
q: 9 is a multiple of 4.
Rewrite the following statements in the form of conditional statements:
A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder.
Translate the following statements in symbolic form:
If x = 7 and y = 4” then x + y = 11.
Identify the quantifiers and write the negation of the following statements:
There exists a number which is equal to its square.
Which of the following is a statement?
The negation of the statement “It is raining and weather is cold.” is ______.
Which of the following sentences are statements? Justify
Every square is a rectangle.
Find the component statements of the following compound statements.
Two lines in a plane either intersect at one point or they are parallel.
Write the component statements of the following compound statements and check whether the compound statement is true or false.
57 is divisible by 2 or 3.
Translate the following statements into symbolic form
2, 3 and 6 are factors of 12
Write down the negation of following compound statements
All real numbers are rationals or irrationals.
Write down the negation of following compound statements
x = 2 and x = 3 are roots of the Quadratic equation x2 – 5x + 6 = 0.
Rewrite the following statements in the form of conditional statements
The square of an odd number is odd.
Rewrite the following statements in the form of conditional statements
You will fail, if you will not study.
Rewrite the following statements in the form of conditional statements
2b = a + c, if a, b and c are in A.P.
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 real number x such that x2 + 1 = 0.
The connective in the statement “2 + 7 > 9 or 2 + 7 < 9” is ______.
The negation of the statement “Rajesh or Rajni lived in Bangalore” is ______.
The negation of the statement “101 is not a multiple of 3” is ______.
Which of the following is the conditional p → q?
