Tag: implications

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Questions Related to implications

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.

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 a dog is barking,then it will not bite"
$(i)$Converse of the statement:If a dog will bite then the dog is barking.
$(ii)$Contrapositive of the statement:If a dog will bite then the dog is not barking.

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

"If a dog is barking,then it will not bite"
$(i)$Converse of the statement:If a dog will not bite then the dog is barking.
$(ii)$Contrapositive of the statement:If a dog will bite then the dog is not barking.

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

Which of the following is true about the converse and contrapositive of the statement
"If two triangles are congruent, then their areas are equal."
(i) Converse of the statement :
If the areas of the two triangles are equal, then the triangles are congruent.
(ii) Contrapositive of the statement:
If the areas of the two triangles are not equal, then the triangles are not congruent.

  1. (i) True (ii) False

  2. (i) False (ii) True

  3. (i) True (ii) True

  4. (i) False (ii) False
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

"If two triangles are congruent, then their areas are equal."
(i) Converse of the statement :
If the areas of the two triangles are equal, then the triangles are congruent.
(ii) Contrapositive of the statement:
If the areas of the two triangles are not equal, then the triangles are not congruent.

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

$p\wedge q$ is logically equivalent to

  1. $\sim(p\rightarrow\sim q)$
  2. $(p\rightarrow\sim q)$
  3. $(\sim p\rightarrow\sim q)$
  4. $(\sim p\rightarrow q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$ $q$ $p\wedge q$ $\sim(p\rightarrow\sim q)$
$T$ $T$ $T$ $T$
$T$ $F$ $T$ $T$
$F$ $F$ $T$ $T$
$F$ $T$ $T$ $T$

$(p\wedge q)\longrightarrow $$\sim(p\rightarrow\sim q)$ is a tautology.

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

The contrapositive of $p \to \left( { \sim q \to  \sim r} \right)$ is 

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

Contraceptive of $a \rightarrow b$ is $\sim b \rightarrow \sim a$


Then

Contraceptive of $p \rightarrow (\sim q \rightarrow \sim r )$

$\equiv \sim (\sim q \rightarrow \sim r) \rightarrow \sim p$

$\equiv \sim (q \wedge \sim r) \rightarrow \sim p [a \rightarrow b \equiv \sim a \wedge b]$

$\equiv (\sim q \vee r) \rightarrow \sim p $  [Demorgas law]

$A$ is correct.