Examples: 1. “We won the match!” The sentence “We won the match!” Is an exclamatory sentence. ∴ It is not a statement. 2. Can u get me cup of tea? This sentence is an interrogative sentence. ∴ It is...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) This sentences is a statement. (ii) Is the earth round?
Answers: (i) This sentences is a statement. The statement is not indicating the correct value, so we can say that value contradicts the sense of the sentence. ∴ It is not a statement. (ii) Is the...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Every rhombus is a square. (ii) x2 + 5|x| + 6 = 0 has no real roots.
Answers: (i) Every rhombus is a square. This sentence is always false, because Rhombuses are not a square. ∴ It is a statement. (ii) x2 + 5|x| + 6 = 0 has no real roots. Let us take, If x>0,...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Are all circles round? (ii) All triangles have three sides.
Answers: (i) Are all circles round? The given sentence is an interrogative sentence. ∴ It is not a statement. (ii) All triangles have three sides. The given sentence is a true declarative sentence....
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Two non-empty sets have always a non-empty intersection. (ii) The cat pussy is black.
Answers: (i) Two non-empty sets have always a non-empty intersection. This sentence is always false, because there are non-empty sets whose intersection is empty. ∴ It is a statement. (ii) The cat...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Listen to me, Ravi! (ii) Every set is a finite set.
Answers: (i) Listen to me, Ravi! The sentence “Listen to me, Ravi! “Is an exclamatory sentence. ∴ It is not a statement. (ii) Every set is a finite set. This sentence is always false, because there...
Check whether the following pair of statements is a negation of each other. Give reasons for your answer. (i) a + b = b + a is true for every real number a and b. (ii) There exist real numbers a and b for which a + b = b + a.
Answer: The negation of the statement: p: a + b = b + a is a true for every real number a and b. ~p: There exist real numbers are ‘a’ and ‘b’ for which a+b ≠ b+a. The statement is not the negation...
Write the negation of the following statement: r: There exists a number x such that 0 < x < 1.
Answer: The negation of the statement: r: There exists a number x such that 0 < x < 1. ~r: For every real number x, either x ≤ 0 or x ≥ 1.
Write the negation of the following statements: (i) p: For every positive real number x, the number (x – 1) is also positive. (ii) q: For every real number x, either x > 1 or x < 1.
Answers: (i) The negation of the statement: 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...
Are the following pairs of statements are a negation of each other: (i) The number x is not a rational number. The number x is not an irrational number. (ii) The number x is not a rational number. The number x is an irrational number.
Answers: (i) “The number x is an irrational number.” The statement “The number x is not a rational number.” It is a negation of the first statement. (ii) “The number x is an irrational number.” The...
(i) Both the diagonals of a rectangle have the same length. (ii) All policemen are thieves.
Answers: (i) The negation of the statement is: “There is at least one rectangle whose both diagonals do not have the same length.” (ii) The negation of the statement is: “No policemen are...
(i) There is a complex number which is not a real number. (ii) I will not go to school.
Answers: (i) The negation of the statement is: It is false that “complex numbers are not a real number.” [Or] “All complex number are real numbers.” (ii) The negation of the statement is: “I will...
(i) All birds sing. (ii) Some even integers are prime.
Answers: (i) The negation of the statement is: It is false that “All birds sing.” [Or] “All birds do not sing.” (ii) The negation of the statement is: It is false that “even integers are prime.”...
Write the negation of the following statement: The sun is cold.
Answer: The negation of the statement is: It is false that “The sun is cold.” [Or] “The sun is not cold.”
Write the negation of the following statement: (i) Ravish is honest. (ii) The earth is round.
Answers: (i) The negation of the statement is: It is false that “Ravish is honest.” [Or] “Ravish is not honest.” (ii) The negation of the statement is: It is false that “The earth is round.” [Or]...
Write the negation of the following statement: (i) Bangalore is the capital of Karnataka. (ii) It rained on July 4, 2005.
Answers: (i) The negation of the statement is: It is false that “Bangalore is the capital of Karnataka.” [Or] “Bangalore is not the capital of Karnataka.” (ii) The negation of the statement is: It...
Determine whether the following compound statements are true or false: (i) Delhi is in India and 2 + 2 = 5 (ii) Delhi is in England and 2 + 2 = 5
Answers: (i) The components of the compound statement are: P: Delhi is in India. Q: 2 + 2 = 5 Both P and Q are false. The compound statement is False. (ii) The components of the compound statement...
Determine whether the following compound statements are true or false: (i) Delhi is in India and 2 + 2 = 4 (ii) Delhi is in England and 2 + 2 = 4
Answers: (i) The components of the compound statement are: P: Delhi is in India. Q: 2 + 2 = 4 Both P and Q are true. The compound statement is True. (ii) The components of the compound statement...
Write the component statements of the following compound statements and check whether the compound statement is true or false: The sand heats up quickly in the sun and does not cool down fast at night.
Answer: The components of the compound statement are: P: The sand heats up quickly in the sun. Q: The sand does not cool down fast at night. P is false and Q is also false then P and Q both are...
Write the component statements of the following compound statements and check whether the compound statement is true or false: (i) Square of an integer is positive or negative. (ii) x = 2 and x = 3 are the roots of the equation 3×2 – x – 10 = 0.
Answers: (i) The components of the compound statement are: P: Square of an integer is positive. Q: Square of an integer is negative. Both P and Q are true. The compound statement is True. (ii) The...
Write the component statements of the following compound statements and check whether the compound statement is true or false: (i) To enter into a public library children need an identification card from the school or a letter from the school authorities. (ii) All rational numbers are real and all real numbers are not complex.
Answers: (i) The components of the compound statement are: P: To get into a public library children need an identity card. Q: To get into a public library children need a letter from the school...
For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer. (i) A lady gives birth to a baby boy or a baby girl. (ii) To apply for a driving license, you should have a ration card or a passport.
Answers: (i) “A lady gives birth to a baby boy or a baby girl.” An exclusive “OR” is used because a lady cannot give birth to a baby who is both a boy and a girl. (ii) “To apply for a driving...
For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer. (i) Students can take Hindi or Sanskrit as their third language. (ii) To entry a country, you need a passport or a voter registration card.
Answers: (i) “Students can take Hindi or Sanskrit as their third language.” An exclusive “OR” is used because a student cannot take both Hindi and Sanskrit as the third language. (ii) “To entry a...
Find the component statements of the following compound statements: (i) All rational numbers are real, and all real numbers are complex. (ii) 25 is a multiple of 5 and 8.
Answers: (i) The components of the compound statement are: P: All rational number is real. Q: All real number are complex. (ii) The components of the compound statement are: P: 25 is multiple of 5....
Find the component statements of the following compound statements: (i) The sky is blue, and the grass is green. (ii) The earth is round, or the sun is cold.
Answers: (i) The components of the compound statement are: P: The sky is blue. Q: The grass is green. (ii) The components of the compound statement are: P: The earth is round. Q: The sun is...
Negate each of the following statements: (i) All the students complete their homework. (ii) There exists a number which is equal to its square.
Answers: (i) The negation of the statement is “Some of the students did not complete their homework.” (ii) The negation of the statement is “For every real number x, x2≠x.”
Write the negation of each of the following statements: (i) For every x ∈ N, x + 3 > 10 (ii) There exists x ∈ N, x + 3 = 10
Answers: (i) The negation of the statement is “There exist x ∈ N, such that x + 3 ≥ 10.” (ii) The negation of the statement is “There exist x ∈ N, such that x + 3 ≠...
Determine the Contrapositive of each of the following statement: If x is an integer and x2 is odd, then x is odd.
Answer: The statement: If x is an integer and x2 is odd, then x is odd. Contrapositive: If x is even, then x2 is even.
Determine the Contrapositive of each of the following statements: (i) It is necessary to be strong in order to be a sailor. (ii) Only if he does not tire will he win.
Answers: (i) Contrapositive: If he is not strong, then he is not a sailor (ii) Contrapositive: If he tries, then he will not win.
Determine the Contrapositive of each of the following statements: (i) If x is less than zero, then x is not positive. (ii) If he has courage he will win.
Answers: (i) Contrapositive: If x is positive, then x is not less than zero. (ii) Contrapositive: If he does not win, then he does not have courage.
Determine the Contrapositive of each of the following statements: (i) It never rains when it is cold. (ii) If Ravish skis, then it snowed.
Answers: (i) Contrapositive: If it rains, then it is not cold. (ii) Contrapositive: If it did not snow, then Ravish will not ski.
Determine the Contrapositive of each of the following statements: (i) If she works, she will earn money. (ii) If it snows, then they do not drive the car.
Answers: (i) Contrapositive: If she does not earn money, then she does not work. (ii) Contrapositive: If then they do not drive the car, then there is no snow.
Determine the Contrapositive of each of the following statements: (i) If Mohan is a poet, then he is poor. (ii) Only if Max studies will he pass the test.
Answers: (i) Contrapositive: If Mohan is not poor, then he is not a poet. (ii) Contrapositive: If Max does not study, then he will not pass the test.
Rewrite each of the following statements in the form “p if and only is q.” (i) r: For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly. (ii) s: If a tumbler is half empty, then it is half full, and if a tumbler is half full, then it is half empty.
Answers: (i) In the form “p if and only is q.” You get an A grade if and only if you do all the homework regularly. (ii) In the form “p if and only is q.” A tumbler is half empty if and only if it...
Rewrite each of the following statements in the form “p if and only is q.” (i) p: If you watch television, then your mind is free, and if your mind is free, then you watch television. (ii) q: If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral is a rectangle, then it is equiangular.
Answers: (i) In the form “p if and only is q.” You watch television if and only if your mind is free. (ii) In the form “p if and only is q.” A quadrilateral is a rectangle if and only if it is...
State the converse and contrapositive of each of the following statement: If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Answer: Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Contrapositive: If the diagonals of a quadrilateral do not bisect each other, then the...
State the converse and contrapositive of each of the following statements: (i) A positive integer is prime only if it has no divisions other than 1 and itself. (ii) If you live in Delhi, then you have winter clothes.
Answers: (i) Converse: If an integer has no divisor other that 1 and itself, then it is prime. Contrapositive: If an integer has some divisor other than 1 and itself, then it is prime. (ii) ...
State the converse and contrapositive of each of the following statements: (i) If it is hot outside, then you feel thirsty. (ii) I go to a beach whenever it is a sunny day.
Answers: (i) Converse: If you feel thirsty, then it is hot outside. Contrapositive: If you do not feel thirsty, then it is not hot outside. (ii) Converse: If I go to a beach, then it is a sunny...
Write each of the following statements in the form “if p, then q.” (i) Whenever it rains, it is cold. (ii) It never rains when it is cold.
Answers: (i) In the form “if p, then q.” If it rains, then it is cold. (ii) In the form “if p, then q.” If it is cold, then it never rains.
Write each of the following statements in the form “if p, then q.” (i) The game is canceled only if it is raining. (ii) It rains only if it is cold.
Answers: (i) In the form “if p, then q.” If it is raining, then the game is canceled. (ii) In the form “if p, then q.” If it rains, then it is cold.
Write each of the following statements in the form “if p, then q.” (i) It is necessary to have a passport to log on to the server. (ii) It is necessary to be rich in order to be happy.
Answers: (i) In the form “if p, then q.” If you log on the server, then you must have a passport. (ii) In the form “if p, then q.” If he is happy, then he is rich.
Write each of the following statements in the form “if p, then q.” (i) You can access the website only if you pay a subscription fee. (ii) There is traffic jam whenever it rains.
Answers: (i) In the form “if p, then q.” If you access the website, then you pay a subscription fee. (ii) In the form “if p, then q.” If it rains, then there is a traffic jam.
Determine whether the argument used to check the validity of the following statement is correct: p: “If x2 is irrational, then x is rational.” The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.
Answer: Argument Used: x2 = π2 is irrational So, x = π is irrational. p: “If x2 is irrational, then x is rational.” Consider, An irrational number given by x = √k [k is a rational number] Squaring...
Which of the following statements are true and which are false? In each case give a valid reason for saying so t: √11 is a rational number.
Answer: t: √11 is a rational number. Square root of prime numbers is irrational numbers. The statement is False.
Which of the following statements are true and which are false? In each case give a valid reason for saying so (i) r: Circle is a particular case of an ellipse. (ii) s: If x and y are integers such that x > y, then – x < – y.
Answers: (i) r: Circle is a particular case of an ellipse. A circle can be an ellipse in a particular case when the circle has equal axes. The statement is true. (ii) s: If x and y are integers such...
Which of the following statements are true and which are false? In each case give a valid reason for saying so (i) p: Each radius of a circle is a chord of the circle. (ii) q: The centre of a circle bisect each chord of the circle.
Answers: (i) p: Each radius of a circle is a chord of the circle. The Radius of the circle is not it chord. The statement is False. (ii) q: The centre of a circle bisect each chord of the circle. A...
By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”
Answer: Consider, A triangle ABC with all angles equal. [Each angle of the triangle is equal to 60] ABC is not an obtuse angle triangle. The statement “p: If all the angles of a triangle are equal,...
Show that the following statement is true “The integer n is even if and only if n2 is even”
Answer: Consider, p: Integer n is even q: If n2 is even Let p be true. Let n = 2k Squaring both the sides, n2 = 4k2 n2 = 2.2k2 n2 is an even number. q is true when p is true. The statement is...
Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x2 is odd, then x is also odd.”
Answer: Consider, q: x is an integer and x2 is odd. r: x is an odd integer. p: if q, then r. Let r be false. x is not an odd integer, then x is an even integer x = (2n) for some integer n x2 = 4n2...
Show that the statement p : “If x is a real number such that x3 + x = 0, then x is 0” is true by method of contradiction
Answer: Method of Contradiction: If possible, let p be false. P is not true -p is true -p (p => r) is true q and –r is true x is a real number such that x3+x = 0and x≠ 0 x =0 and x≠0 This is a...
Show that the statement p : “If x is a real number such that x3 + x = 0, then x is 0” is true by (i) Direct method (ii) method of Contrapositive
Answers: (i) Direct Method: Consider, q: x is a real number such that x3 + x=0. r: x is 0. If q, then r. Let q be true. Then, x is a real number such that x3 + x = 0 x is a real number such that...
Check whether the following statement is true or not: (i) p: If x and y are odd integers, then x + y is an even integer. (ii) q : if x, y are integer such that xy is even, then at least one of x and y is an even integer.
Answers: (i) p: If x and y are odd integers, then x + y is an even integer. Conisder, p: x and y are odd integers. q: x + y is an even integer If p, then q. Let p be true. [x and y are odd integers]...
Check the validity of the following statement: r: 60 is a multiple of 3 or 5.
Answer: r: 60 is a multiple of 3 or 5. We know that 60 is a multiple of 3 as well as 5. Hence, the statement is true.
Check the validity of the following statements: (i) p: 100 is a multiple of 4 and 5. (ii) q: 125 is a multiple of 5 and 7.
Answers: (i) p: 100 is a multiple of 4 and 5. We know that 100 is a multiple of 4 as well as 5. Hence, the statement is true. (ii) q: 125 is a multiple of 5 and 7 We know that 125 is a multiple of 5...
For each of the following compound statements first identify the connecting words and then break it into component statements. (i) All rational numbers are real and all real numbers are not complex. (ii) Square of an integer is positive or negative.
(I) In this sentence 'and' is the associating word The part proclamations are as per the following (a) All normal numbers are genuine (b) All genuine numbers are not perplexing (ii) In this...