Tag: discrete mathematics

Questions Related to discrete mathematics

The only statement among the following taht is a tautology is -

logical equivalence mathematical logic discrete mathematics business maths maths

  1. $A\wedge (A\vee B)$

  2. $A\vee (A\wedge B)$

  3. $[A\wedge (A\rightarrow B)]\rightarrow B$

  4. $B\rightarrow [A\wedge (A\rightarrow B)]$

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Correct Option: A

The contrapositive of the statement "if  $2 ^ { 2 } = 5 ,$  then  $1$  get first class" is

logical equivalence mathematical logic discrete mathematics business maths maths

  1. If I do not get a first class, then $2 ^ { 2 } = 5$

  2. If I do not get a first class, then $2 ^ { 2 } \neq 5$

  3. If I get a first class, then $2 ^ { 2 } = 5$

  4. If I get a first class, then $2 ^ { 3 } = 5$

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Correct Option: B
Explanation:

$P:{ 2 }^{ 2 }=5$

$q:I$ get first class
the contrapositive of $p\rightarrow q$ is $\sim q\rightarrow \sim p$. Hence the answer is if $I$ do not get a first class, then ${ 2 }^{ 2 }\neq 5$
Correct Answer : Option B.

The proposition $( P \Longrightarrow \sim p) ^ (\sim p \Longrightarrow P)$ is 

logical equivalence mathematical logic discrete mathematics business maths maths

  1. Contingency

  2. Neither Tautology nor contradiction

  3. contradiction

  4. Tautology

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Correct Option: A

Which of  the following is logically equivalent to : $\sim \left[\sim p\rightarrow q\right]$

logical equivalence mathematical logic discrete mathematics business maths maths

  1. $p\vee\sim q$

  2. $\sim p\wedge q$

  3. $\sim p\vee q$

  4. $\sim p\wedge \sim q$

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Correct Option: A

The statement  $\sim ( p \wedge q ) \vee q$

logical equivalence mathematical logic discrete mathematics business maths maths

  1. is a tautology

  2. is equivalent to $( p \wedge q ) \vee ( - q )$

  3. is equivalent to $p \vee q$

  4. is a contradiction

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Correct Option: A

The simplicity $ \sim(p \rightarrow q) \longleftrightarrow(\sim p \vee \sim q) $ is

logical equivalence mathematical logic discrete mathematics business maths maths

  1. tautology

  2. contradiction

  3. neither t nor e

  4. None of these.

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Correct Option: A

Consider :
Statement - I :$(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy.
Statement - II :$(p\rightarrow q)\leftrightarrow (\sim q\rightarrow \sim p)$ is a tautology.

logical equivalence mathematical logic discrete mathematics business maths maths

  1. Statement - I is true: Statement - II is true: Statement - II is a correct explanation for Statement - I.

  2. Statement - I is true: Statement - II is true: Statement - II is not a correct explanation for Statement - I.

  3. Statement - I is true; Statement - II is false.

  4. Statement - I is false; Statement - II is true.

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Correct Option: A

The statement (p ^ q) ^ (-pv - q) is _______________.

logical equivalence mathematical logic discrete mathematics business maths maths

  1. a tautology

  2. a contradiction

  3. a contingency

  4. neither a tautology nor a contradiction

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Correct Option: A

Statement $(p\wedge  q) \rightarrow p$ is

logical equivalence mathematical logic discrete mathematics business maths maths

  1. a tautology.

  2. a contradiction.

  3. neither a tautology nor a contradiction.

  4. none of these.

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Correct Option: A

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

logical equivalence mathematical logic discrete mathematics business maths maths

  1. a tautology

  2. a contradiction

  3. neither a tautology nor a contradiction

  4. None of these

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Correct Option: C
Explanation:

When $p$ and $q$ both are true then 

$(p \rightarrow  q) and (\sim p \vee \sim q)$ both are false

i.e. $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is true when $p$ and $q$ both are false then  

$\sim (p \rightarrow  q)$ is false and $(\sim p \vee \sim q)$ is true

i.e. $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is false

Hence $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is neither tautology nor contradiction