Tag: principle of mathematical induction

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Questions Related to principle of mathematical induction

Multiple choice business maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Which of the following is logically equivalent to $\displaystyle \sim \left (\sim p\rightarrow q\right )$?

  1. $\displaystyle p\wedge q$
  2. $\displaystyle p\wedge \sim q$
  3. $\displaystyle \sim p\wedge q$
  4. $\displaystyle \sim p\wedge \sim q$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
p  q  $\sim p$  $\sim q$  $\sim p \rightarrow q$  $\sim (\sim p \rightarrow q)$  $p \wedge q$  $p \wedge \sim q$   $\sim p \wedge q$   $\sim p \wedge \sim q$  
T  T  F  F  T  F  T  F  F  F 
T F  F  T  T  F  F  T  F  F 
F T  T  F  T  F  F  F  T  F 
F F  T  T  F  T  F  F  F  T 

The values in column 6 and column 10 are same.

Hence, option D is correct.

Multiple choice business maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The dual of the following statement "Reena is healthy and Meena is beautiful" is

  1. Reena is not beaufiful and Meena is not healthy.

  2. Reena is not beautiful or Meena is not healthy.

  3. Reena is not healthy or Meena is not beautiful.

  4. None of these.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $p$ denote the statement "Reena is healthy" 


and $q$ denote the statement "Meena is beautiful"

Now the given statement is $p\wedge q$

Now the Dual of this statement will be obtained by replacing $\vee$ by 

$\wedge$ and $\wedge$ by $\vee$ and inversing the true value of the statement.

So the Dual of $p\wedge q$ will be $\sim p\vee \sim q$

The statement $\sim p$ will be "Reena is not healthy"

The statement $\sim q$ will be "Meena is not beautiful"

So the dual statement will be $\sim p\vee \sim q$ or "Reena is not healthy or Meena is not beautiful."

Multiple choice business maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The statement "If $2^2 = 5$ then I get first class" is logically equivalent to

  1. $2^2 = 5$ and I do not get first class
  2. $2^2 = 5$ or I do not get first class
  3. $2^2 \neq 5$ or I get first class

  4. None of these.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

There can be two cases
$2^{2}=5$ $\rightarrow$ first class.
$2^{2}\neq 5$\rightarrow not a first class.
Hence logically equivalent statement will be 
$2^{2}=5$ or $2^{2}\neq 5$ but $2^{2}=5$ statement is equivalent to getting first class.
Hence
First class or $2^{2}\neq 5$.

Multiple choice business maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The statement "If $2^2 = 5$ then I get first class" is logically equivalent to

  1. $2^2 = 5$ and I donot get first class
  2. $2^2 = 5$ or I do not get first class
  3. $2^2 \neq 5$ or I get first class

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Obviously, ${ 2 }^{ 2 }\neq 5$, then the statement will be ${ 2 }^{ 2 }\neq 5$ or $I$ get first class.

Multiple choice business maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Logically equivalent statement to $p \leftrightarrow  q$ is

  1. $(p \rightarrow q)\wedge (q \rightarrow p)$
  2. $(p \wedge q)\vee (q \rightarrow p)$
  3. $(p \wedge q)\rightarrow (q \vee p)$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 $p$  $q$  $p\leftrightarrow q$
 T  T  T
 T  F  F
 F  T  F
 F  F  T
 $p$  $q$  $p\rightarrow q$  $q\rightarrow p$ $\left( p\longrightarrow q \right) \wedge \left( q\longrightarrow p \right) $ $p\wedge q$  $\left( p\wedge q \right) \vee \left( q\longrightarrow p \right) $ $q\vee p$  $\left( p\wedge q \right) \longrightarrow \left( q\vee p \right) $ 
 T  T  T  T  T  T  T  T  T
T   F  F  T  F  F  T  T  T
 F  T  T  F  F  F  F  T  T
 F F   T  T  T  F  T  F  T
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Which one of the statement gives the same meaning of statement
If you watch television, then your mind is free and if your mind is free then you watch television

  1. You watch television if and only if your mind is free.

  2. You watch television and your mind is free.

  3. You watch television or your mind is free.

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
"You watch television and your mind is free".
The above statement gives or suits for the same meaning of the structure given because it is logically correct.
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Which of the following is NOT true for any two statements $p$ and $q$?

  1. $\sim[p\vee (\sim q)]=(\sim p)\wedge q$
  2. $\sim(p\vee q)=(\sim p)\vee (\sim q)$
  3. $q\wedge \sim q$ is a contradiction
  4. $\sim (p\wedge (\sim p))$ is a tautology
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$p$ and $q$ are two statements.
$A) LHS = \sim [pv (\sim q)]$
By De morgon's laws
$\sim(pr (\sim q))= \sim pnq$
$\therefore (A) $ is true .

$B) \sim(p v q) = (\sim p) \vee (\sim q)$
According to demorgon's laws, this is false.
$\because \sim (p \vee q) = (\sim p)\wedge (\sim q)$. 
$\therefore (B)$ is false.

$C) q \wedge \sim  q$ is a contradiction because $'q'$ and $\sim q$ are opposite statements i.e, cannot be there at the same time.

$D) \sim (p \wedge (\sim p))$
$p \wedge (\sim p)$ is a contradiction, which is evident from option $(C)$. $\therefore $ opposite of a contradiction is a tautology .
$\therefore [B]$ is wrong.
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

If p and q are two statements, then statement $p\Rightarrow q\wedge \sim q$.

  1. Tautology

  2. Contradiction

  3. Neither tautology nor contradiction

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the statement p implies (q and not q), the consequent (q and not q) is always false (a contradiction). An implication with a false consequent and a variable antecedent has a truth value that depends on p, making it neither a tautology nor a contradiction.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The statement $\sim (p \leftrightarrow \sim q)$ is

  1. Equivalent to $\sim p \leftrightarrow q$

  2. A tautology

  3. A fallacy
  4. Equivalent to $p \leftrightarrow q$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The biconditional p <-> q means both have the same truth value, while p <-> not q means they have opposite truth values. Negating a biconditional that equates p to not q flips it back to equating p directly to q.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The proposition $\left( {p \wedge q} \right) \Rightarrow p$ is 

  1. neither tautology nor contradiction

  2. A tautology

  3. A contradiction

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The proposition (p and q) implies p means that whenever both p and q are true, p must be true, which is always correct by definition of conjunction and implication. Thus, it is a tautology.