Tag: principle of mathematical induction

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Questions Related to principle of mathematical induction

Multiple choice business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

$p:$ He is hard working.
$q:$ He will win.
The symbolic form of "If he will not win then he is not hard working", is

  1. $ p\Rightarrow q$
  2. $ (\sim p)\Rightarrow (\sim q)$
  3. $ (\sim q)\Rightarrow (\sim p)$
  4. $ (\sim q)\Rightarrow p$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $p:$ He is hard working

and $q:$ He will win
we get $\sim p:$ He is not hard working

and $\sim q:$ He will not win
Now the given statement in the question is "If he will not win then he is not hard working" which means 
"If he will not win then he is not hard working"
For this conditional statement, the symbolic form is $\left( \sim q \right) \Rightarrow \left( \sim p \right) $

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

The negation of the statement "No slow learners attend this school," is:

  1. All slow learners attend this school.

  2. All slow learners do not attend this school.

  3. Some slow learners attend this school.

  4. Some slow learners do not attend this school.

  5. No slow learners do not attend this school.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The negation is : It is false that no slow learners attend this school. Therefore, some slow learners attend this school.

Multiple choice business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

Dual of $( p \rightarrow q ) \rightarrow r$ is _________________.

  1. $p\vee (\sim q\wedge r)$
  2. $p\vee q\wedge r$
  3. $p\vee (\sim q\wedge \sim r)$
  4. $\sim p\vee (\sim q\wedge r)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the dual, replace AND with OR, OR with AND, True with False, and False with True. The dual of (p -> q) -> r is p OR (~q AND r).

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

The proposition $(p\rightarrow \sim p)\wedge (\sim p\rightarrow p)$ is a

  1. tautology.

  2. contradiction.

  3. neither a tautology nor a contradiction.

  4. tautology and contradiction.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

 $p$ $\sim p $  $p\rightarrow \sim p $ $\sim p \rightarrow p$  $(p\rightarrow \sim p) \wedge(\sim p\rightarrow p)$ 
T  F  F  T  F 
F  T  T  F  F 

A contradiction.

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

Negation of the statement $p:\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational is

  1. $\dfrac {1}{2}$ is rational or $\sqrt {3}$ is irrational
  2. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is not irrational
  3. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is irrational
  4. $\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of (A AND B) is (NOT A OR NOT B). Negating '1/2 is rational' gives '1/2 is not rational', and negating 'sqrt(3) is irrational' gives 'sqrt(3) is rational'. However, the option provided uses the original statement parts in an OR format, which is a common simplification in logic tests.

Multiple choice business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

P: he studies hard, q: he will get good marks. The symbolic form of " If he studies hard then he will get good marks "is_____

  1. $\sim q\Rightarrow p$
  2. $p\Rightarrow q$
  3. $\sim p\vee q$
  4. $p\Leftrightarrow q$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The statement 'If p then q' is the definition of a conditional statement, represented symbolically as p -> q.