Mathematical Reasoning(11th Grade > Mathematics ) Questions and Answers

Question 1.


For any two statements p and q, the statement ∼(p ∨ q)∨(∼p ∧ q) is equivalent to


  1.     ∼p
  2.     p
  3.     q
  4.     ∼q
Explanation:-
Answer: Option A. -> ∼p
:
A

∼(p ∨ q)∨(∼p ∧ q)=(∼p ∧∼q) ∨ (∼ p ∧ q)=(∼ p ∧(q ∨∼q))=∼p 


Question 2.


The statement p⇒∼q is equivalent to


  1.     q⇒p
  2.     ∼q⇒p
  3.     ∼q ∨∼p
  4.     ∼p ∨ q
Explanation:-
Answer: Option C. -> ∼q ∨∼p
:
C

p⇒q is false only when p is true and q is false.
p⇒∼q is false when p is true and ~q is false i.e. when both p and q are true.


It can be seen that ∼q ∨∼p is false only when p and q are both true. Hence it is the equivalent of p⇒q


Question 3.


If p: The earth is round, q : 3 + 4 =7, then ∼p ∨∼q is 


  1.     It is not that the earth is round or 3+4 =7
  2.     The earth is round and 3+4 =7
  3.     It is not that the earth is round or it is not that 3+4 =7
  4.     The earth is round or it is not that 3+4 =7
Explanation:-
Answer: Option C. -> It is not that the earth is round or it is not that 3+4 =7
:
C

p: the earth is round
q: 3+4 =7
~p: It is not that the earth is round
~q: It is not that 3+4 =7
∼p ∨∼q: It is not that the earth is round or it is not that 3+4 =7


Question 4.


Identify the false statement.


  1.     ∼(p ∨∼q)=∼p ∧ q
  2.     ∼(p ∨ q)=∼p ∨∼q
  3.     (p ∧ q)∧∼p is a contradiction
  4.     (p ∨ q)∨∼p is a tautology
Explanation:-
Answer: Option B. -> ∼(p ∨ q)=∼p ∨∼q
:
B

∼(p ∧ q)=∼p ∧∼q
Hence option B is false.


Question 5.


Let P be the statement 'Ravi races' and let Q be the statement 'Ravi wins'. Then the verbal translation of ∼(P ∨∼Q) is


  1.     Ravi does not race and Ravi does not win
  2.     It is not true that Ravi races and that Ravi does not win
  3.     Ravi does not race or Ravi wins
  4.     It is not true that Ravi races or that Ravi does not win
Explanation:-
Answer: Option D. -> It is not true that Ravi races or that Ravi does not win
:
D

P: Ravi races
Q: Ravi wins
~Q: Ravi does not win
P ∨∼Q: Ravi races or Ravi does not win
∼(P ∨∼Q): It is not true that Ravi races or that Ravi does not win


Question 6.


The negation of the statement (p ∨∼q)∧q is


  1.     (∼p ∨ q)∧∼q
  2.     (p ∧∼q)∨q
  3.     (∼p ∧ q)∨∼q
  4.     (p ∧∼q)∨∼q
Explanation:-
Answer: Option C. -> (∼p ∧ q)∨∼q
:
C

∼((p ∨∼q)∧ q)=∼(p ∨∼q)∨∼q=(∼p ∧ q)∨∼q


Question 7.


Find the negation of the following statement. 'All cats scratch'.


  1.     No cat scratches
  2.     All that scratch are not cats
  3.     There exists a cat that does not scratch
  4.     None of these
Explanation:-
Answer: Option C. -> There exists a cat that does not scratch
:
C

The negation of the statement would be 'not all cats scratch' or there exists at a cat that does not scratch.


Question 8.


Identify the correct statement(s).


  1.     'There exists' is the existential quantifier
  2.     'For all' is the universal quantifier
  3.     'For all' is the existential quantifier
  4.     'There exists' is the universal quantifier
Explanation:-
Answer: Option A. -> 'There exists' is the existential quantifier
:
A and B

'There exists' is the existential quantifier and 'for all' is the universal quantifier


Question 9.


Negation of the statement 'If we control population, we prosper' is 


  1.     If we do not control population, we prosper.
  2.     If we control population, we do not prosper
  3.     We control population and we do not prosper
  4.     We do not control population and we prosper
Explanation:-
Answer: Option C. -> We control population and we do not prosper
:
C

The negation of p⇒q is p ∧∼q 
p: We control population
q: We prosper
p ∧∼q: We control population and we do not prosper


Question 10.


The statement ∼(p⇒q) is equivalent to


  1.     p ∧∼q
  2.     ∼p ∧ q
  3.     p ∧ q
  4.     ∼p ∧∼q
Explanation:-
Answer: Option A. -> p ∧∼q
:
A

∼(p⇒q) is true only when p is true and q is false. Hence it is logically equivalent to p ∧∼q


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