Tag: different forms of theoretical statements

Questions Related to different forms of theoretical statements

Without using truth , whether
$\left[ {p\Delta \left( { \sim q\Delta r} \right)} \right]V\left[ { \sim r\Delta  \sim q\Delta p} \right] \equiv p$

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

  1. True

  2. False

Show answer
Correct Option: B

Solve it:-
$\left( {p \to q} \right) \to [\left( { \sim p \to q} \right) \to q]$

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

  1. Tautology

  2. Contradiction

  3. Contingent

  4. Not statement

Show answer
Correct Option: A
Explanation:
$Y=\left( p\longrightarrow q \right) \longrightarrow \left[ \left( \sim p\longrightarrow q \right) \longrightarrow q \right]$
Method : TRUTH TABLE [  ALWAYS PREFERABLE]

 $p$  $q$  $p\longrightarrow q$  $\left( \sim p\longrightarrow q \right) $  $\left[ \left( \sim p\longrightarrow q \right) \longrightarrow q \right]$  $Y$
 $1$  $0$  $0$  $1$  $1$ $1$ 
 $1$  $1$  $1$  $1$  $1$  $1$
 $0$  $0$  $1$  $0$  $1$  $1$
 $0$  $1$  $1$  $1$  $1$  $1$
As the result is always TRUE $\left(i.e. 1\right)$;
$\left( p\longrightarrow q \right)\longrightarrow \left[ \left( \sim p\longrightarrow q \right) \longrightarrow q \right]$ is Tautology.

A. Tautology



















Let $p$ and $q$ be two propositions given by
$p$ : The sky is blue.
$q$ : The milk is white.
Then $p\wedge q$ will be

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

  1. The sky if blue or milk is white

  2. The sky is blue and milk is white

  3. The sky is white and milk is blue

  4. If the sky is blue then milk is white

Show answer
Correct Option: B
Explanation:

$p \wedge q$ means statement p and q
=> The sky is blue and  milk is white.

Let $p$ and $q$ be two propositions. Then the contrapositive the implication $p\rightarrow q$

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

  1. $\sim q\rightarrow \sim p$

  2. $\sim p\rightarrow \sim q$

  3. $q\rightarrow p$

  4. $p\leftrightarrow q$

Show answer
Correct Option: A
Explanation:

the contrapositive of $p\to q$ is $\sim q\to \sim p$

Consider the following statements 
$p$:you want to success
$q$:you will find way,
then the negation of $\sim (p\vee q)$ is

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

  1. you want of success and you find a way

  2. you want of success and you do not find a way

  3. if you do not want to succeed then you will find a way

  4. if you want of success then you cannot find a way

Show answer
Correct Option: A

Which of the following statements is a tautology

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

  1. $\left( { \sim p \vee q} \right) - \left( {p \vee \sim q} \right)$

  2. $\left( { \sim p \vee \sim q} \right) \to p \vee q$

  3. $\left( {p \vee \sim q} \right) \wedge \left( {p \vee q} \right)$

  4. $\left( { \sim p \vee \sim q} \right) \vee \left( {p \vee q} \right)$

Show answer
Correct Option: C

Which of the following is a logical statement?

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

  1. Open the door

  2. What an intelligent student!

  3. Are you going to Delhi

  4. All prime numbers are odd numbers

Show answer
Correct Option: D
Explanation:

The above $3$ statements are basic statements.

But the $4$ statement is a logical statement.
All prime numbers are odd numbers.

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

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

  1. neither tautology nor contradiction

  2. A tautology

  3. A contradiction

  4. Cannot be determined

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Correct Option: B

The statement $p \to (q \to p)$ is equivalent to 

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

  1. $p \to q$

  2. $p \to (q \vee p)$

  3. $p \to (q \to p)$

  4. $p \to (q \wedge p)$

Show answer
Correct Option: B

Which of the following is correct?

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

  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. $~(p \leftrightarrow q) \equiv (p \rightarrow q) \wedge (q \rightarrow p)$

Show answer
Correct Option: D
Explanation:


$~(p \leftrightarrow q) \equiv (p \rightarrow q) \wedge (q \rightarrow p)$ is true, we show it by truth table using boolean expression.

1.$p\rightarrow q$=min(1,1+q-p)
2.$p\wedge q$=min(p,q)
3.$p\leftrightarrow q$=1-|p-q|

Now we draw or make truth table using these operations
L.H.S  

 p  q $p\leftrightarrow q$ 
 1 1  1 
1  0  0 


R.H.S 

p q  $p\rightarrow q$  $q\rightarrow p$   $(p \rightarrow q) \wedge (q \rightarrow p)$
1  1  1  1  1
1  1  0 1  0 

L.H.S =R.H.S

$~(p \leftrightarrow q) \equiv (p \rightarrow q) \wedge (q \rightarrow p)$