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

Logically equivalent statement to $p \leftrightarrow  q$ is

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

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

  1. Tautology

  2. Contradiction

  3. Neither tautology nor contradiction

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the statement p implies (q and not q), the consequent (q and not q) is always false (a contradiction). An implication with a false consequent and a variable antecedent has a truth value that depends on p, making it neither a tautology nor a contradiction.

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

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

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

  2. A tautology

  3. A fallacy
  4. Equivalent to $p \leftrightarrow q$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The biconditional p <-> q means both have the same truth value, while p <-> not q means they have opposite truth values. Negating a biconditional that equates p to not q flips it back to equating p directly to q.

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

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

  1. neither tautology nor contradiction

  2. A tautology

  3. A contradiction

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The proposition (p and q) implies p means that whenever both p and q are true, p must be true, which is always correct by definition of conjunction and implication. Thus, it is a tautology.

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

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

  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. $B\rightarrow [A\wedge (A\vee B)]$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

[A ^ (A -> B)] -> B is Modus Ponens, which is a tautology. If A is true and (A -> B) is true, then B must be true.

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

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

  1. $145^{o}$
  2. $150^{o}$
  3. $155^{o}$
  4. $160^{o}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

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.

  1. $(i)$True and $(ii)$False
  2. $(i)$False and $(ii)$True
  3. $(i)$True and $(ii)$True
  4. $(i)$False and $(ii)$False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.