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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Statement : if a natural number is even, then is square is also even. write this statement in 5 different forms of the same meaning. |
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Answer» (1).(i) a natural number is even imlies that its square is even. (ii) a natural number is even only if its square is even. (iii) for a natural number to be even it is necessary that its square is even. (iv) for the square of a natural number to be even it is sufficient the number is even. (v) if the square of a natural number is not even, then the natural number is not even. |
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| 2. |
Identify the quantifier in the following statements and write thenegation of the statements.(i) There exists anumber which is equal to its square.(ii) For every realnumber x, x is less than `x" "+" "1`.(iii) There exists acapital for e |
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Answer» (i) Quantifier : there exists negation : there does not exist a number which is equal to its square. (ii) Quantifier: for every. Negation : there exists a real number x such that x is more than `(x+1)` . (iii) Quantifier : there exists. Negation: there exists a state in india which does not have a capital. |
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| 3. |
Identify the quantifers used in the following statements and write the negation of the statements : (i) there exists a number which is equal to its cube. (ii) for all states in india, there is a capital in india. (iii) there exists a man whose age is 150 years. (iv) all students are of 25 years or more. |
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Answer» (i) there exists : negation : there dones not exist a number which is equal to its cube. (ii) for all : negation : there is a state in india which has no capital. (iii) There exists: Negation : There does not exist a man whose age is 150 years. (iv) for all : negation: no student is of 25 years or more. |
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| 4. |
Write the negation of the following statements:(i) p: For every real number `x ,``x^2> x`.(ii) q: There exists a rational number x such that `x^2=2`.(iii) r: All birds have wings.(iv) s: All students study mathematics at the element |
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Answer» (i) `~p : `For every real number `x, x^2 < x` (ii)`~q : `There exists a rational number `x` such that `x^2 !=2`. (iii)`~r: `All birds do not have wings. (iv) `~s: `All students do not study mathematics at the elementary level. |
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| 5. |
Write the negation of the following statements:(i) p : For every positive real number x, the number `x - 1`is also positive.(ii) q : All cats scratch.(iii) r : For every real number x, either `x > 1`or `x < 1`.(iv) s : There exist |
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Answer» (i) there exists a positive real number x such that `x-1` is not positive. (ii) there exists at least one cat which does not scratch. (iii) there exists a real number `x` such that neither `xgt1` nor `xlt1`. (iv) For every real number `x,x le0`or `xge1`. |
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| 6. |
Prove that `sqrt(2)`is an irrational number. |
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Answer» Let statement `p:sqrt(2)` is an irrational number. Let if possible p is false. `therefore sqrt(2)` is a rational number. `rArr.sqrt(2)=(a)/(b)` where a and b are prime integers and `bne0`. `rArr 2=(a^2)/(b^2)` `rArr a^2=2b^2` `rArra^2` is divisible by 2. `rArra` is divisible by 2. Let `a=2c` where c is an interger. `rArr a^2=4c^2` `rArr 2b^2=4c^2" "(because a^2=2b^2)` `rArr b^2=2c^2` `rArrb^2` is divisible by 2. `rArr` b is divisible by 2 . Now a and b both are divisible by 2. which contradiction our assumption that a and b are prime. `therefore sqrt(2)` is rational number is false. `rArr sqrt(2)` is an irrational number. |
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| 7. |
Given below are two statementsp : 25 is a multiple of5.q : 25 is a multiple of8.Write the compound statements connecting these two statements with"And" and "Or". In both cases check the validity of thecompound statement. |
| Answer» Correct Answer - 1 | |
| 8. |
Show that the statement p: If x is a real number such that `x^3+4x=0`. then x is 0 is true by(i) direct method, (ii) method of contradiction,(iii) method of contrapositive. |
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Answer» Given statement : if x is a real number such that `x^3+4x=0` , then ` x=0` Comonent p:x is a real number such that `x^3+4x=0` Component `q:x=0` (i) Direct method : `because x ` is a real number and `x^3+4x=0` `therefore x(x^2+4)=0` `rArrx=0` or ` x^2+4=0` But `x^2+4` cannot be zero. `therefore x=0` (ii) Method of contradication : Let x is a real number and `x^3+4x=0` let `xne0` Now `x(x^2+4)=0` `rArr(x^2+4)=0` `rArrx=-` or `x^2+4=0` But `x^2+4` cannot be zero. `therefore x=0` (iii) Method of contrapositive : let `xne0` `rArrx` is a real number and `xne0` `therefore x^2gt0` `rArrx^2+4gt0` `becausexne0` and `x^2+4gt0` `rArrxne0` and `x^2+4ne0` `rArrx(x^2+4)ne0` `rArrx^3+4xne0` therefore, `xne 0 rArrx^3+4xne0` `rArrx^3+4x=0rArr x=0` |
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| 9. |
Verify by the method of contradiction. `p:sqrt(7)`is irrational. |
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Answer» Let us assume `p:sqrt7` is a rational. `=>sqrt7 = a/b` It means, ,`a` and `b` have no common factor. Also, `a^2/b^2 = 7=> a^2 = 7b^2->(1)` Above equation means, `a^2 ` has a factor of `7`. So, `a` will also have a factor of `7`.We can also write it as, `a = 7c->(2)` where `c` is another real number. From (1) and (2), `49c^2 = 7b^2=> b^2 = 7c^2` Above means, that `b` is also having a factor of `7`. But, earlier we proved `a` and `b` have no common factor. So, our assumption is incorrect that `sqrt7` is a rational number. It means, `sqrt7` is an irrational number. |
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| 10. |
Check the validity of the statements given below by the method given against it.(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).(ii) q: If n is a real number with `n > 3`, then `n^2>9 |
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Answer» (i) True (ii) True |
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| 11. |
The negation of the statement '101 is not mulitple of 3' isA. 101 is a multiple 3B. 101 is a multiple of 2C. 101 is an odd numberD. 101 is an greater than 5 |
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Answer» Correct Answer - A Let p: 101 is not a muliple of 3. ~ p: 101 is a multiple of 3. |
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| 12. |
Write the negation of the following statements:(i) Chennai is the capital of Tamil Nadu,(ii) `sqrt(2)`is not a complex number(iii) All triangles are not equilateral triangle.(iv) The number 2 is greater than 7.(v) Every natural number is |
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Answer» (i) Chennai is not the capital of tamil nadu. (ii) `sqrt(2)` is a complex number. (iii) all triangles are not equilateral triangle. (iv) the number 2 is greater than 7. (iv) every natural number is an integer. |
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| 13. |
State whether the following statements are negation to each other ? (i) p: ram is a good boy. q : ram is not a good boy. (ii) p: `sqrt(5)` is a rational number. q : `sqrt(5)` is an irational number. (iii) p: australia is a continent. q: australia is not a continent. (iv) p: a multiple of 2 is 16. q : a multiple of 2 is 12. |
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Answer» (i) yes (ii) yes (iii) yes (iv) No. |
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| 14. |
The statement `~(pharr~ q)`is(1) equivalent to`pharrq`(2) equivalent to`~ pharrq`(3) atautology(4) a fallacy |
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Answer» Here, we will represent `T` for True and `F` for False. When `p=T` and `q = T`, `~(p harr ~q) = T`, `~p harr q = F`, `p harr q = T` When `p=T` and `q = F`, `~(p harr ~q) = F`, `~p harr q = T`, `p harr q = F` When `p=F` and `q = T`, `~(p harr ~q) = F`, `~p harr q = T`, `p harr q = F` When `p=F` and `q = F`, `~(p harr ~q) = T`, `~p harr q = F`, `p harr q = T` `:. ~(p harr ~q) ` is equivalent to `p harr q`. Also, it is not a tautology as it contains false. It is not a fallacy as it contains true also. So, option `1` is the correct option. |
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| 15. |
Write each of the following statements in the form "if-then"(i) You get a job impliesthat your credentials are good.(ii) The Banana treeswill bloom if it stays warm for a month.(iii) A quadrilateral is aparallelogram if its diagonals bisect |
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Answer» (i) if you get a job , then your credentials are good (ii) if the banana tree stays warm for a month, then it will bloom (iii) it diagonals of a quadrilateral bisect each other, then it is a parallelogram. (iv) if you get `A^(+)` grade in the class, then you do all the exercise in the book. |
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| 16. |
The connective in the statement 'Earth revolves round the Sun and Moon is a satellite of earth ' isA. orB. EarthC. SunD. and |
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Answer» Correct Answer - D Connective word is |
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| 17. |
Which of the following is the conditional `p to q`?A. q is sufficient for pB. p is necessary for qC. p only if qD. if q then p |
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Answer» Correct Answer - C `p to q` is same as p only if q. |
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| 18. |
The negative of the statement 'The product of 3 and 4 is 9' isA. it is false that the product of 3 and 4 is 9B. the product of 3 and 4 is 12C. the product of 3 and 4 is not 12D. it is false that the product of 3 and 4 is not 9 |
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Answer» Correct Answer - A The negative of the above statement is it is false that a natural number is not greater than zero. |
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| 19. |
Which of the following is not a negative of 'A nature number is greater than zero'A. A natural number is not greater than zeroB. It is false that a natural number is gerater than zeroC. It is false that a natural number is not greater than zeroD. None of the above |
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Answer» Correct Answer - C The false negative of the given statement is 'it is false that a natural number is not greater than zero. |
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| 20. |
Check whether the following sentences are statements. Give reasons for your answer.(i) 8 is less than 6.(ii) Every set is a finite set.(iii) The sun is a star.(iv) Mathematics is fun.(v) There is no rain without clouds.(vi) How far is Chennai from here? |
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Answer» (i) false statement (ii) false statement (iii) true statement (iv) not a statement (v) true statement (vi) not a statement. |
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| 21. |
For each of the following statements, determine whether an inclusive Or or exclusive Or is used. Give reasons for your answer.(i) To enter a country, you need a passport or a voter registration card.(ii) The school is closed if it is a holiday or a Sunday.(iii) Two lines intersect at a point or are parallel.(iv) Students can take French or Sanskrit as their third language. |
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Answer» (i) inclusive (ii) inclusive (iii) exclusive (iv) exclusive |
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| 22. |
Which of the following statement is a conjunction ?A. Ram and Shyam are friendsB. Both Ram and Shyam are tallC. Both Ram and Shyam are enemiesD. None of the above |
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Answer» Correct Answer - D 0 |
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| 23. |
Which of the following sentences are statements? Justify your answer: (i) there are 40 days in a month. (ii) the sum of 7 and 10 is 17.(iii) the diagonals of a rectangle are equal. (iv) the heat prouduced from the fire. |
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Answer» (i) no month has 40 days, so the given sentences is false and therefore it is a statement. (ii) the given sentence is true, therefore it is a statement. (iii) the diagonals of envery rectangle are equal, so it is a true sentence and therefore it is a statement. (iv) the heat always produced by the fire, so the sentence is true, therefore it is a statement. |
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| 24. |
The contrapositive of the statement 'If 7 is greater than 5, then 8 is greater than 6' isA. If 8 is greater than 6, then 7 is greater than 5B. If 8 is not greater than 6 then 7 greater than 5C. If 8 is not greater than 6, then 7 is greater tha 5D. If 8 is greater than 6, then 7 is not greater than 5 |
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Answer» Correct Answer - C Let p: 7 is gretaer than 5. and q:8 is greater than 6. `therefore p to q` ~ p: 7 is not greater than 5. ~q : 8 is not greater than 6. `(~q) to (~p)` i.e., If is 8 is not greater than 6, then 7 is not greater than 5. |
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| 25. |
By giving a counter example, show that the following statements are not true.(i) p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.(ii) q: The equation `x^2-1=0`does not have a root lying between 0 |
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Answer» (i) here p: all angles of a triangle are equal. q : triangle is obtuse angled triangle. all angles of `Delta` are equal then each angle will be `60^@`. Therefore , p is true then q is false. `therefore` given statement is not true. (ii) we prove it by a counter example. for htis we need such value which is the root of the equation and it does not lie between 0 and 2. `x=-1` is a root of this equation and it does not lies between 0 and 2. therefore, given statement is not true. |
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| 26. |
Write the component statements of the following compound statements and then check whether the statements is true it is isoceles. (i) if a triangle is equilateral then it is isosceles . (ii) if a and b are two natural numbers, then `a+b` is also a natural number. |
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Answer» (i) component statements are : p: a triangle is equilateral. q : a triangle is isosceles. `because` every equilateral triangle is isoceles. `because` Given statement is true. (ii) Component statements are : p : a and b are natural numbers. q:`a+b` is a natural number. `because` the sum of two natural numbers is always a natural number : `therefore` given statements is true. |
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| 27. |
Write each of the statements in the form "if p, then q"(i) p : It isnecessary to have a password to log on to the server.(ii) q : There istraffic jam whenever it rains.(iii) r : You canaccess the website only if you pay a subscription fee. |
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Answer» (i) if you log on to the server, then you have a password. (ii) it is rains, then is traffic jam. (iii) if you can access the website then you pay a subscription fee. |
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| 28. |
Write the contra positive of the following statement:(i) If a number is divisible by 9, then it is divisible by 3.(ii) If you are born in India, then you are a citizen of India.(iii) If a triangle is equilateral, it is isosceles. |
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Answer» (i) if a no is not divisible by 3 then its not divisible by 9 (ii) if you are not a citizen of India then you are not born in india (iii) if a triangle is not isosceles then it is not equilateral |
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| 29. |
Identify the type of Or used in the following statements and cheek whether the statements are true or false:(i) `sqrt(2)`is a rational number or an irrational number.(ii) To enter into a public library children need an identity card from the school authorities.(iii) A rectangle is a quadrilateral or a 5-sided polygon. |
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Answer» (i)`sqrt2`is a rational number or an irrational number. a)`sqrt2`is a rational number.` -> False` b) `sqrt2`is an irrational number.` -> True` Out of these two statements, only one statement can be true. So, type of this statement is `Exclusive` `OR`. As, one of the statement is true, so this statement is true. (ii) To enter into a public library children need an identity card from the school or a letter from school authorities. a) To enter into a public library children need an identity card from the school. b) To enter into a public library children need a letter from school authorities. Both of these satatements can be true. So, type of this statement is `Inclusive` `OR`. Also, both of these statements are false as to enter into a public library children do not need any permission from school. (iii) A rectangle is a quadrilateral or a 5-sided polygon. a)A rectangle is a quadrilateral. `->True`. b)A rectangle is a 5-sided polygon.`-> False` Out of these two statements, only one statement can be true. So, type of this statement is `Exclusive` `OR`. As, one of the statement is true, so this statement is true. |
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| 30. |
State the converse and contrapositive of each of the followingstatements:(i) p : A positiveinteger is prime only if it has no divisors other than 1 and itself.(ii) q : I go to abeach whenever it is a sunny day.(iii) r : If it is hotouts |
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Answer» (i) Given statement : a positive integer is prime only if it has no divisors other than 1 and itself. Converse: if a positive integer has no divisors other than 1 and itself, then it is a prime. Contrapositive : if positive integer has divisors other than 1 and itself, then it is not prime. (ii) Given statement : I go to a beach whenever it is a sunny day. Converse : if I go to beach, it is a sunny day. Contrapositive : if I do not go to beach, then it is not a sunny day. (iii) given statement: if it is hot outside, then you feel thirsty. Converse: if you feel thirsty, it is hot outside, Contrapositive : if you do not feel thirsty. then it is not hot outside. |
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| 31. |
Write the conctrapositive and converse of the followig statements. (i) if x is a prime number, then x is odd. (ii) if the two lines are parallel, then they do not intersect in the same plane. (iii) something is cold imlies that it has low temperature. (iv) you cannot comprehend geometry if you do not know how to reson duductively. (v) x is an even number implies that x is divisible by 4. |
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Answer» (i) Given statement : if x is a prime number, then x is odd. Contrapositive : if x is not odd then x is not a prime number. Converse: if a number x is odd then x is a prime number . (ii) Given statement : if two lines are parallel,then they do not intersect in the same plane. Contrapositive : if two lines interset in a plane , then they are not parallel . converse: if two lines do not intersect in a plane, then they are parallel. (iii) Given statement : something is cold implies that it has low temperature. Contrapositive : if something is not at low temperature, then it is not cold. Converse: if something is at low temperature , then it is cold. (iv) Given statement : you cannot comprehend geometry if you do not know how to reason deductively. Contrapositive : if you know how to reason deductively , then you can comprehend geometry. Converse: if you do not know how to reason deductively , then you cannot comprehend geometry. (v) Given statement : x is an number implies that x is divisibile by 4. Contrapositive : x is not divisible by 4, then x is not an even number . Converse: If x is divisible 4, then x is an even number. |
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| 32. |
The converse of the statement 'If sum is not shining, then sky is filled with clouds ' isA. if Sky is filled with clouds, then the Sun is not sharingB. If Sun is shining, then sky is fielld with cloundsC. If sky is clear, then Sun is shiningD. |
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Answer» Correct Answer - A Let p: Sum is not shining. and q : sky is filled with clouds. Converse of the above statement `pto q` is `q to q` If sky is filled with clouds, then the Sun is not shinig. |
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| 33. |
The contrapositive of the statement 'if p, then q ', isA. if q , then pB. if p, then ~qC. if ~q. then ~pD. if ~p, then ~q |
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Answer» Correct Answer - C `p to q` If p, then q Contrapositive of the statement `p to q` is `(~q) to (~p)`. |
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| 34. |
Find the component statements of the following compound statementsand check whether they are true or false.(i) Number 3 is prime orit is odd.(ii) All integers arepositive or negative.(iii) 100 is divisible by3, 11 and 5. |
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Answer» (i) component statements p: number 3 is prime. q: number 3 is odd. both components statements are true. (ii) component statements p: all integeres are positive. q: all integers are negative. both component statements are false. (iii) component statements p: 100 is divisible by 3 q: 100 is divisible by 11 r: 100 is divisible by 5. here the component p and q are false and r is true. |
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| 35. |
Let S be a non-empty subset of R. Consider the following statement:P: There is a rational number `x in S`such that `x"">""0`.Which of the following statements is the negation of the statement P ?There is no rational number `x in S`such that `xlt=0`(9)Every rationalnumber `x in S`satisfies `xlt=0`(18)`x in S`and `xlt=0=>x`(27) is notrationalThere is arational number `x in S`such that `xlt=0`(36) |
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Answer» `x in S` `x > 0` `x in S; x <= 0` option 2 is correct |
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| 36. |
Thenegation of `~ svv(~ r^^s)`is equivalent to :(1) `s^^~ r`(2)`s^^(r^^~ s)`(3)`svv(rvv~ s)`(4)`s^^r` |
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Answer» `~(-s vv (~ r ^^ s))` `~(~s) ^^ ~ ( ~ r ^^ s)` `s ^^ ( ~ (~r) vv ~ s)` `s ^^ ( r vv ~ s)` `( s ^^ r) vv ( s ^^ ~ s) ` `s ^^ r vv f` `s ^^ r` Answer |
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