Tag: different forms of theoretical statements

Questions Related to different forms of theoretical statements

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.

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.

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

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

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

  3. $(i)$False $(ii)$False

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

Show answer
Correct Option: D
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.

Write the dual of the following statement:
(p$\vee$ q)$\wedge$ T

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

  1. (p$\wedge$ q) $\vee$ T

  2. (p$\wedge$ q) $\vee$ F

  3. (p$\vee$ q) $\vee$ F

  4. (p$\vee$ q) $\vee$ T

Show answer
Correct Option: B
Explanation:

To obtain the dual of a formula , replace ∧ with V, T with F and vice versa

So, B is correct option.

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.

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

  1. (i) True (ii) False

  2. (i) False (ii) True

  3. (i) True (ii) True

  4. (i) False (ii) False

Show answer
Correct Option: C
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.

Identify the Law of Logic: $p \wedge q \equiv q \wedge p$

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

  1. Idempotent Law

  2. Commutative Law

  3. Associative Law

  4. Conditional Law

Show answer
Correct Option: B
Explanation:
Given logic 
$p\wedge q\equiv q \wedge p$ 
 It is commutative law 
according to commutative law the order does not matter
we can also check it by truth table 

$p\wedge q$ is logically equivalent to

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

  1. $\sim(p\rightarrow\sim q)$

  2. $(p\rightarrow\sim q)$

  3. $(\sim p\rightarrow\sim q)$

  4. $(\sim p\rightarrow q)$

Show answer
Correct Option: A
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.

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

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

  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. $p \wedge \left( {q \vee r} \right)$

Show answer
Correct Option: A
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.