Tag: proofs in mathematics

Quizzes (2) Questions (121)

Questions Related to proofs in mathematics

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

Which of the following is correct?

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

  4. none of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Clearly, the statements $p \vee q$ and $p\wedge q$ cannot be equivalent as they one operator means "OR" and the other operator means "AND".

$p$ $q$ $p\rightarrow q$ $q\rightarrow p$
T T T T
T F F T
F T T F
F F T T

Option B is also incorrect.

$p$ $q$ $p\rightarrow q$ $p\wedge q$
T T T T
T F F F
F T T F
F F T F


Hence, option C is also incorrect.

Option D is also incorrect as
$p \leftrightarrow q=(p\rightarrow q)\wedge (q\rightarrow p)$


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

Which of the following statement are NOT logically equivalent?

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

We make an option wise check for this.

Option A: $\sim \left( p\vee \sim q \right) \quad and\quad \left( \sim p\wedge q \right) $
By application of Demorgan's Law on $\sim \left( p\vee \sim q \right) $ we get, $\sim \left( p\wedge q \right) $ 
So this option is logically equivalent.

Option B: $\sim \left( p\longrightarrow q \right) \quad and\quad \left( p\wedge \sim q \right) $
Again by application Conditional Disjunction rule, we see that this option is also logically equivalent.

Option C: $\left( p\longrightarrow q \right) \quad and\quad \left( \sim q\longrightarrow \sim p \right) $
This is again true by Contrapositive tautology.

Option D:$\left( p\longrightarrow q \right) \quad and\quad \left( \sim p\wedge q \right) $
This is not logically equivalent. 

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

The statement $\sim (p\rightarrow \sim q)$ is equivalence to ___________.

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

$\sim\left({p} \rightarrow \sim{q} \right)$

We know that,
               $\sim\left({p} \rightarrow {q} \right)={p}\wedge\sim{q}$
          $\Rightarrow\sim\left({p}\rightarrow\sim{q}\right)=\sim{p}\wedge\sim\left(\sim{q}\right)$
                                    $=\sim{p}\wedge{q}$
Hence, $\left(\sim{p}\wedge{q}\right)$ is the correct answer.


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

Which of the following is always true?

  1. $\sim(p\rightarrow q) \equiv \sim p \wedge q$
  2. $\sim(p\vee q) \equiv \sim p \vee \sim q$
  3. $\sim (p \implies q ) \equiv (p \land \sim q )$
  4. $\sim(p \wedge q) \equiv \sim p \wedge \sim q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$p \implies q \equiv \sim p \lor q  $
$\therefore \sim (p \implies q ) \equiv \sim (\sim p \lor q )$
$\therefore \sim (p \implies q ) \equiv (p \land \sim q )$

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

Which of the following is/are false?

  1. $p\rightarrow q\equiv\sim p\rightarrow\sim q$
  2. $\sim(p \rightarrow\sim q)\equiv\sim p\wedge q$
  3. $\sim(\sim p\rightarrow\sim q)\equiv\sim p\wedge q$
  4. $\sim (p\leftrightarrow q) \equiv(\sim(p\rightarrow q))\wedge\sim(q\rightarrow p)$
Reveal answer Fill a bubble to check yourself
A,B,D Correct answer
Explanation

We know that:
$p\rightarrow q \equiv \sim q\rightarrow \sim p$    {By logical equivalences }    
Hence $A$ is false


Now for option $B$
$\sim (p \ \rightarrow \ \sim q)$ $\equiv$ $\sim (\sim p\vee \sim q)=p\wedge q$   [By logical Equivalences ]
Hence $B$ is false

Now for option $C$
$\sim (\sim p\rightarrow \sim q)$ $\equiv \sim (p  \vee \sim q) $  $\equiv \sim p\wedge q$  [By Logical Equivalences]
Hence $C$ is true

Now for option $D$
$\sim (p\leftrightarrow q)$ $\equiv \sim ((p\rightarrow q)\wedge (q\rightarrow p))$ $\equiv \sim (p\rightarrow q)\vee \sim (q\rightarrow p)$
Hence $D$ is false                        [By logical Equivalences]

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.