Tag: proofs in mathematics

Questions Related to proofs in mathematics

Logically equivalent statement to $p \leftrightarrow  q$ is

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

  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

Show answer
Correct Option: A
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

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

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

  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

Show answer
Correct Option: B
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.

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

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

  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

Show answer
Correct Option: B
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.

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

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

  1. Tautology

  2. Contradiction

  3. Neither tautology nor contradiction

  4. None of these

Show answer
Correct Option: A

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

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

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

  2. A tautology

  3. A fallacy

  4. Equivalent to $p \leftrightarrow q$

Show answer
Correct Option: C

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

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

Show answer
Correct Option: C

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

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

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

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

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

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

Show answer
Correct Option: C

A clock is started at noon. By 10 min past 5, the hour hand has turned through

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

  1. $145^{o}$

  2. $150^{o}$

  3. $155^{o}$

  4. $160^{o}$

Show answer
Correct Option: C
Explanation:

Angle traced by hour hand in 12 h = $360^{o}$

Angle traced by hour hand in 5 h 10 min i.e., $\dfrac{31}{6} h$ $\implies (\dfrac{360}{12} \times \dfrac{31}{6})^{o}$ = $155^{o}$

A symbol $(\alpha)$ is used to represent 10 flowers. Number of symbols to be drawn to show 60 flowers is

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

  1. $6\alpha$

  2. $12\alpha$

  3. $20\alpha$

  4. $24\alpha$

Show answer
Correct Option: A
Explanation:

A symbol α is used to represent 10 flowers.
60 flowers = 6 ×  10 = 6α

Write the converse and contrapositive of the statement
"If it rains then they cancel school."
$(i)$Converse of the statement :
If they cancel school then it rains.
$(ii)$Contrapositive of the statement:
If it does not rain then they do not cancel school.

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

  1. $(i)$True and $(ii)$False

  2. $(i)$False and $(ii)$True

  3. $(i)$True and $(ii)$True

  4. $(i)$False and $(ii)$False

Show answer
Correct Option: A
Explanation:

"If it rains then they cancel school."
$(i)$Converse of the statement :
If they cancel school then it rains.
$(ii)$Contrapositive of the statement:
If they do not cancel school then it does not rain.