Tag: implications

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

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

  1. Tautology

  2. Contradiction

  3. Contingent

  4. Not statement
Reveal answer Fill a bubble to check yourself
A Correct answer
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



















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

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

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

Negation of $(\sim p\rightarrow q)$ is ________________.

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

The negation of an implication (p -> q) is (p AND NOT q). Applying this to (~p -> q), we get (~p AND NOT q), which simplifies to (~p AND ~q).

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

$( p \wedge q ) \vee ( \sim p \wedge q ) \vee ( \sim q \wedge r ) =? $

  1. $q \vee r$
  2. $q \wedge r$
  3. $q \rightarrow r$

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

Applying Boolean algebra or logic laws to the expression (p and q) or (not p and q) or (not q and r), we can factor out q from the first two terms to get q, which then combines with the third term to simplify to q or r.

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

If $p$ is false, $q$ is true, then which of the following is/are false?

  1. $\sim (p\Rightarrow q)$
  2. $\sim p$
  3. $\sim p\Rightarrow q$
  4. $\sim q$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation
 $p$  $q$  $p\Rightarrow q$ $\sim \left( p\Rightarrow q \right) $  $\sim p$  $\sim q$   $\sim p\Rightarrow q$
 F  T  T  F  T  F  T

Here we see that $\sim \left( p\Rightarrow q \right) $  and $\sim q$  are false

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

$p$: He is hard working.
$q$: He is intelligent.
Then $ \sim q\Rightarrow\sim p$, represents

  1. If he is hard working, then he is not intelligent.

  2. If he is not hard working, then he is intelligent.

  3. If he is not intelligent, then he is not had working.

  4. If he is not intelligent, then he is hard working.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

p:she is hardworking
q:she is intelligent

~p:she is not hardworking
~q:she is not intelligent

~q=>~p means She is not intelligent implies she is not hardworking
Hence, Option C