JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Question Bank

done Self Evaluation Test - Mathematical Reasoning Total Questions - 50

• question_answer1)
  If \[S(p,q,r)=(\tilde{\ }p)\vee [\tilde{\ }(q\wedge r)]\] is a compound statement, then \[S(\tilde{\ }p,\tilde{\ }q,\tilde{\ }r)\] is
  A)
  \[\tilde{\ }S(p,q,r)\] done clear
  B)
  \[S(p,q,r)\] done clear
  C)
  \[p\vee (q\wedge r)\] done clear
  D)
  \[p\vee (q\vee r)\] done clear
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• question_answer2)
  Which of the following is the inverse of the proposition: ?if a number is a prime then it is odd.??
  A)
  If a number is not a prime then it is odd done clear
  B)
  If a number is not a prime then it is not odd done clear
  C)
  If a number is not odd then it is not a prime done clear
  D)
  If a number is not odd then it is a prime? done clear
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• question_answer3)
  Which of the following is a statement?
  A)
  Open the door. done clear
  B)
  Do your homework. done clear
  C)
  Switch on the fan. done clear
  D)
  Two plus two is four. done clear
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• question_answer4)
  Truth value of the statement ?It is false that \[3+3=33\] Or \[1+2=12'\] is
  A)
  T done clear
  B)
  F done clear
  C)
  Both T and F done clear
  D)
  54 done clear
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• question_answer5)
  Which of the following statement is a contradiction?
  A)
  \[(\tilde{\ }p\vee \tilde{\ }q)\vee (p\vee \tilde{\ }q)\] done clear
  B)
  \[(p\to q)\vee (p\wedge \tilde{\ }q)\] done clear
  C)
  \[(\tilde{\ }p\wedge q)\wedge (\tilde{\ }q)\] done clear
  D)
  \[(\tilde{\ }p\wedge q)\vee (\tilde{\ }q)\] done clear
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• question_answer6)
  Which of the following is not a statement?
  A)
  Please do me a favour done clear
  B)
  2 is an even integer done clear
  C)
  \[2+1=3\] done clear
  D)
  The number 17 is prime done clear
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• question_answer7)
  Negation of 'Paris in France and London is in England'' is
  A)
  Paris is in England and London is in France done clear
  B)
  Paris is not in France or London is not in England done clear
  C)
  Paris is in England or London is in France done clear
  D)
  None of these done clear
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• question_answer8)
  The contrapositive of \[p\to (\tilde{\ }q\to \tilde{\ }r)\] is-
  A)
  \[(\tilde{\ }q\wedge r)\to \tilde{\ }p\] done clear
  B)
  \[(q\to r)\to \tilde{\ }p\] done clear
  C)
  \[(q\vee \tilde{\ }r)\to \tilde{\ }p\] done clear
  D)
  None of these done clear
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• question_answer9)
  If p : It is snowing, q : I am cold, then the compound statement 'it is snowing and it is not that I am cold'? is given by
  A)
  \[p\wedge (\tilde{\ }q)\] done clear
  B)
  \[p\wedge q\] done clear
  C)
  \[(\tilde{\ }p)\wedge q\] done clear
  D)
  \[(\tilde{\ }p)\wedge (\tilde{\ }q)\] done clear
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• question_answer10)
  Consider the statement p: 'New Delhi is city'. Which of the following is not negation of p?
  A)
  New Delhi is not a city done clear
  B)
  It is false that New Delhi is a city done clear
  C)
  It is not the case that New Delhi is a city done clear
  D)
  None of these   done clear
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• question_answer11)
  Which of the following is not a proposition
  A)
  \[\sqrt{3}\] is a prime done clear
  B)
  \[\sqrt{2}\] is irrational done clear
  C)
  Mathematics is interesting done clear
  D)
  5 is an even integer done clear
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• question_answer12)
  Which of the following is always true?
  A)
  \[(\tilde{\ }p\vee \tilde{\ }q)\equiv (p\wedge q)\] done clear
  B)
  \[(p\to q)\equiv (\tilde{\ }q\to \tilde{\ }p)\] done clear
  C)
  \[\tilde{\ }(p\to \tilde{\ }q)\equiv (p\wedge \tilde{\ }q)\] done clear
  D)
  \[\tilde{\ }(p\leftrightarrow q)\equiv (p\to q)\to (q\to p)\] done clear
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• question_answer13)
  The statement \[p\to (q\to p)\]is equivalent to
  A)
  \[p\to (p\to q)\] done clear
  B)
  \[p\to (p\vee q)\] done clear
  C)
  \[p\to (p\wedge q)\] done clear
  D)
  \[p\to (p\leftrightarrow q)\] done clear
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• question_answer14)
  The inverse of the statement \[(p\wedge \tilde{\ }q)\to r\] is
  A)
  \[\tilde{\ }(p\vee \tilde{\ }q)\to \tilde{\ }r\] done clear
  B)
  \[(\tilde{\ }p\wedge q)\to \tilde{\ }r\] done clear
  C)
  \[(\tilde{\ }p\vee q)\to \tilde{\ }r\] done clear
  D)
  None of these done clear
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• question_answer15)
  Identify the false statements
  A)
  \[\tilde{\ }[p\vee (\tilde{\ }q)]\equiv (\tilde{\ }p)\vee q\] done clear
  B)
  \[[p\vee q]\vee (\tilde{\ }p)\] is a tautology done clear
  C)
  \[[p\wedge q)\wedge (\tilde{\ }p)\] is a contradiction done clear
  D)
  \[\tilde{\ }[p\vee q]\equiv (\tilde{\ }p)\vee (\tilde{\ }q)\] done clear
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• question_answer16)
  Let p and q be any two logical statements and \[r:p\to (\tilde{\ }p\vee q).\] If r has a truth value F, then the truth values of p and q are respectively:
  A)
  F, F done clear
  B)
  T, T done clear
  C)
  T, F done clear
  D)
  F, T done clear
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• question_answer17)
  If :p Raju is tall and q: Raju is intelligent, then the symbolic statement \[\tilde{\ }p\vee q\] means
  A)
  Raju is not tall or he is intelligent. done clear
  B)
  Raju is tall or he is intelligent done clear
  C)
  Raju is not tall and he is intelligent done clear
  D)
  Raju is not tall implies he is intelligent done clear
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• question_answer18)
  Negation of the conditional: 'If it rains, I shall go to school'? is
  A)
  It rains and I shall go to school done clear
  B)
  It runs and I shall not go to school done clear
  C)
  It does not rains and I shall go to school done clear
  D)
  None of these done clear
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• question_answer19)
  If \[p\Rightarrow (q\vee r)\] is false, then the truth values of \[p,q,r\] are respectively
  A)
  \[T,F,F\] done clear
  B)
  \[F,F,F\] done clear
  C)
  \[F,T,T\] done clear
  D)
  \[T,T,F\] done clear
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• question_answer20)
  Consider the two statements p: he is intelligent and Q: He is strong. Then the symbolic form of the statement ?It is not true that he is either intelligent or strong?? is
  A)
  \[\tilde{\ }p\vee Q\] done clear
  B)
  \[\tilde{\ }P\wedge \tilde{\ }Q\] done clear
  C)
  \[\tilde{\ }P\wedge Q\] done clear
  D)
  \[\tilde{\ }(P\vee Q)\] done clear
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• question_answer21)
  If p: Ashok works hard  q: Ashok gets good grade The verbal form for \[(\tilde{\ }p\to q)\] is
  A)
  If Ashok works hard then gets good grade done clear
  B)
  If Ashok does not work hard then he gets good grade done clear
  C)
  If Ashok does not work hard then he does not get good grade done clear
  D)
  Ashok works hard if and only if he gets grade done clear
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• question_answer22)
  For integers m and n, both greater than 1, consider three following three statements:

  P: m divides n

  Q: m divides \[{{n}^{2}}\]

  R: m is prime, then

  A)
  \[Q\wedge R\to P\] done clear
  B)
  \[P\wedge Q\to R\] done clear
  C)
  \[Q\to R\] done clear
  D)
  \[Q\to P\] done clear
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• question_answer23)
  If \[p\Rightarrow (\tilde{\ }p\,\vee \,q)\] is false, the truth values of p and q are respectively
  A)
  \[F,T\] done clear
  B)
  \[F,F\] done clear
  C)
  \[T,T\] done clear
  D)
  \[T,F\] done clear
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• question_answer24)
  The negation of the statement \[(p\wedge q)\to (\tilde{\ }p\vee r)\]is
  A)
  \[(p\wedge q)\vee (p\vee \tilde{\ }r)\] done clear
  B)
  \[(p\wedge q)\vee (p\wedge \tilde{\ }r)\] done clear
  C)
  \[(p\wedge q)\wedge (p\wedge \tilde{\ }r)\] done clear
  D)
  \[p\vee q\] done clear
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• question_answer25)
  Let p, q and r be any three logical statements. Which of the following is true?
  A)
  \[\tilde{\ }[p\wedge (\tilde{\ }q)]\equiv (\tilde{\ }p)\wedge q\] done clear
  B)
  \[\tilde{\ }[(p\vee q)\wedge (\tilde{\ }r)\equiv (\tilde{\ }p)\vee (\tilde{\ }q)\vee (\tilde{\ }r)\] done clear
  C)
  \[\tilde{\ }[p\vee (\tilde{\ }q)]\equiv (\tilde{\ }p)\wedge q\] done clear
  D)
  \[\tilde{\ }[p\vee (\tilde{\ }q)]\equiv (\tilde{\ }p)\wedge \tilde{\ }q\] done clear
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• question_answer26)
  \[(p\wedge \tilde{\ }q)\wedge (\tilde{\ }p\wedge q)\] is
  A)
  A tautology done clear
  B)
  A contradiction done clear
  C)
  Both a tautology and a contradiction done clear
  D)
  Neither a tautology nor a contradiction done clear
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• question_answer27)
  Consider the two statements p: He is intelligent and Q: he is strong. Then the symbolic form of the statement ?it is not true that he is either intelligent or strong?? is
  A)
  \[\tilde{\ }P\vee Q\] done clear
  B)
  \[\tilde{\ }P\wedge \tilde{\ }Q\] done clear
  C)
  \[\tilde{\ }P\wedge Q\] done clear
  D)
  \[\tilde{\ }(P\vee Q)\] done clear
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• question_answer28)
  If p is false and q is true, then
  A)
  \[p\wedge q\] is true done clear
  B)
  \[p\vee \tilde{\ }q\] is true done clear
  C)
  \[q\wedge p\] is true done clear
  D)
  \[p\Rightarrow q\] is true done clear
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• question_answer29)
  The propositions \[(p\Rightarrow \tilde{\ }p)\wedge (\tilde{\ }p\Rightarrow p)\] is a
  A)
  Tautology and contradiction done clear
  B)
  Neither tautology nor contradiction done clear
  C)
  Contradiction done clear
  D)
  Tautology done clear
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• question_answer30)
  Which of the following is not logically equivalent to the proposition: 'A real number is either rational or irrational.'?
  A)
  If a number is neither rational n nor irrational then it is not real done clear
  B)
  If a number is not a rational or not an irrational, then it is not real done clear
  C)
  If a number is not real, then it is neither rational nor irrational done clear
  D)
  If a number is real, then it is rational or irrational. done clear
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• question_answer31)
  The negation of the statement ?A circle is an ellipse?? is
  A)
  An ellipse is a circle done clear
  B)
  An ellipse is not a circle done clear
  C)
  A circle is not an ellipse done clear
  D)
  A circle is an ellipse done clear
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• question_answer32)
  Which of the following is false?
  A)
  \[p\vee \tilde{\ }p\] is a tautology done clear
  B)
  \[\tilde{\ }(\tilde{\ }p)\leftrightarrow p\] is a tautology done clear
  C)
  \[p\wedge \tilde{\ }p\] is a contradiction done clear
  D)
  \[((p\wedge q)\to q)\to p\] is a tautology done clear
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• question_answer33)
  Which of the following is always true?
  A)
  \[(\tilde{\ }p\Rightarrow q)=\tilde{\ }q\Rightarrow \tilde{\ }p\] done clear
  B)
  \[(\tilde{\ }p\vee q)\equiv \vee p\vee \tilde{\ }q\] done clear
  C)
  \[\tilde{\ }(p\Rightarrow q)\equiv p\wedge \tilde{\ }q\] done clear
  D)
  \[\tilde{\ }(\,p\,\vee q)\equiv \tilde{\ }p\wedge \tilde{\ }q\] done clear
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• question_answer34)
  The contrapositive of the inverse of \[p\Rightarrow \tilde{\ }q\] is
  A)
  \[\tilde{\ }q\Rightarrow p\] done clear
  B)
  \[p\Rightarrow q\] done clear
  C)
  \[\tilde{\ }q\Rightarrow \tilde{\ }p\] done clear
  D)
  \[\tilde{\ }p\Rightarrow \tilde{\ }q\] done clear
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• question_answer35)
  In the truth table for the statement \[(p\to q)\leftrightarrow (\tilde{\ }p\vee q),\] the last column has the truth value in the following order is
  A)
  \[TTFF\] done clear
  B)
  \[FFFF\] done clear
  C)
  \[TTTT\] done clear
  D)
  \[FTFT\]   done clear
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• question_answer36)
  If p: 4 is an even prime number, q: 6 is a divisor of 12 and r: the HCF of 4 and 6 is 2, then which one of the following is true?
  A)
  \[(p\wedge q)\] done clear
  B)
  \[(p\vee q)\wedge \tilde{\ }r\] done clear
  C)
  \[\tilde{\ }(q\wedge r)p\] done clear
  D)
  \[\tilde{\ }p\vee (q\wedge r)\] done clear
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• question_answer37)
  Which of the following is a contradiction?
  A)
  \[(p\wedge q)\wedge \tilde{\ }(p\vee q)\] done clear
  B)
  \[p\vee (-p\wedge q)\] done clear
  C)
  \[(p\Rightarrow q)\Rightarrow p\] done clear
  D)
  None of these done clear
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• question_answer38)
  Consider the following statements

  p: A tumbler is half empty.

  q: A tumbler is half full.

  The, the combination form of 'p if and only if q'? is

   
  A)
  A tumbler is half empty and half full done clear
  B)
  A tumbler is half empty if and only if it is hatful done clear
  C)
  Both (a) and (b) done clear
  D)
  None of the above done clear
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• question_answer39)
  The contrapositive of \[(p\vee q)\Rightarrow r\] is
  A)
  \[r\Rightarrow (p\vee q)\] done clear
  B)
  \[\tilde{\ }r\Rightarrow (p\vee q)\] done clear
  C)
  \[\tilde{\ }r\Rightarrow \tilde{\ }p\wedge \tilde{\ }q\] done clear
  D)
  \[p\Rightarrow (q\vee r)\] done clear
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• question_answer40)
  The negation of \[(p\vee \tilde{\ }q)\wedge q\] is
  A)
  \[(\tilde{\ }p\vee q)\wedge \tilde{\ }q\] done clear
  B)
  \[(p\wedge \tilde{\ }q)\vee q\] done clear
  C)
  \[(\tilde{\ }p\wedge q)\vee \tilde{\ }q\] done clear
  D)
  \[(p\wedge \tilde{\ }q)\vee \tilde{\ }q\] done clear
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• question_answer41)
  Let p be the proposition: Mathematics is a interesting and let q be the propositions that Mathematics is difficult, then the symbol \[p\wedge q\]means
  A)
  Mathematics is interesting ipllies that Mathematics is difficult done clear
  B)
  Mathematics is interesting impels and is implied by Mathematics is difficult done clear
  C)
  Mathematics is interesting and Mathematics is difficult done clear
  D)
  Mathematics is interesting or Mathematics is difficult done clear
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• question_answer42)
  Which of the following is true?
  A)
  \[p\Rightarrow q\equiv \tilde{\ }p\Rightarrow \tilde{\ }q\] done clear
  B)
  \[\tilde{\ }(p\Rightarrow \tilde{\ }q)\equiv \tilde{\ }p\wedge q\] done clear
  C)
  \[\tilde{\ }(\tilde{\ }p\Rightarrow \tilde{\ }q)\equiv \tilde{\ }p\wedge q\] done clear
  D)
  \[\tilde{\ }(\tilde{\ }p\Leftrightarrow q)\equiv [\tilde{\ }(p\Rightarrow q)\wedge \tilde{\ }(q\Rightarrow p)]\] done clear
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• question_answer43)
  The contrapositive of the statement, ?If I do not secure good marks then I cannot go for engineering?, is
  A)
  If I secure good marks, then I go for engineering done clear
  B)
  If I go for engineering then I secure good marks done clear
  C)
  If I cannot go for engineering then I do not secure good marks done clear
  D)
  None done clear
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• question_answer44)
  \[\tilde{\ }(p\Rightarrow q)\Leftrightarrow \tilde{\ }p\vee \tilde{\ }q\] is
  A)
  A tautology done clear
  B)
  A contradiction done clear
  C)
  Neither a tautology nor a contradiction done clear
  D)
  Cannot come to any conclusion done clear
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• question_answer45)
  If p is nay statement, then which of the following is a tautology?
  A)
  \[p\wedge f\] done clear
  B)
  \[p\vee f\] done clear
  C)
  \[p\vee (\tilde{\ }p)\] done clear
  D)
  \[p\wedge t\] done clear
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• question_answer46)
  The false statement of the following is
  A)
  \[p\wedge (\tilde{\ }p)\] is a contradiction done clear
  B)
  \[(p\Rightarrow q)\Leftrightarrow (\tilde{\ }q\Rightarrow \tilde{\ }p)\] is a contradiction done clear
  C)
  \[\tilde{\ }(\tilde{\ }p)\Leftrightarrow p\] is a tautology done clear
  D)
  \[p\vee (\tilde{\ }p)\Leftrightarrow p\] is a tautology done clear
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• question_answer47)
  Negation of the proposition: If we control population growth, we prosper
  A)
  If we do not control population growth, we prosper done clear
  B)
  If we control population growth, we do not prosper done clear
  C)
  We control population but we do not prosper done clear
  D)
  We do not control population, but we prosper. done clear
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• question_answer48)
  The inverse of the statement, if x is zero then we cannot divide by x? is
  A)
  If we cannot divide by x, then x is zero done clear
  B)
  If we cannot divide by x, then x is not zero done clear
  C)
  If x is not zero then we divide by x done clear
  D)
  None. done clear
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• question_answer49)
  The statement 'If \[{{2}^{2}}=5\] then I get first class'? is logically equivalent to
  A)
  \[{{2}^{2}}=5\] and I do not get first class done clear
  B)
  \[{{2}^{2}}=5\] or I do not get first class done clear
  C)
  \[{{2}^{2}}\ne 5\] or I get first class done clear
  D)
  None of these done clear
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• question_answer50)
  If p and q are two statement then \[(p\leftrightarrow \tilde{\ }q)\] is true when-
  A)
  p and q both are true done clear
  B)
  p and q both are false done clear
  C)
  p is false and q is true done clear
  D)
  None of these done clear
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