Tag: proofs in mathematics

Questions Related to proofs in mathematics

Let $p$ and $q$ be two propositions. Then the contrapositive the implication $p\rightarrow q$

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

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

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

  3. $q\rightarrow p$

  4. $p\leftrightarrow q$

Show answer
Correct Option: A
Explanation:

the contrapositive of $p\to q$ is $\sim q\to \sim p$

Negation of $(\sim p\rightarrow q)$ is ________________.

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $\sim { p }{ \wedge }\sim q$

  2. $\sim \left( p\vee q \right) \vee \left( p\vee \left( \sim p \right) \right) $

  3. $\sim \left( p\vee q \right) \wedge \left( p\vee \left( \sim p \right) \right) $

  4. $\left( \sim p\vee q \right) \wedge \left( p\vee \sim q \right) $

Show answer
Correct Option: A

$p \wedge ( q \vee \sim p ) =?$

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $p \vee q$

  2. $p \wedge q$

  3. $p \rightarrow  q$

  4. none of these

Show answer
Correct Option: B

$( p \wedge q ) \vee ( \sim p \wedge q ) \vee ( \sim q \wedge r ) =? $

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $q \vee r$

  2. $q \wedge r$

  3. $q \rightarrow r$

  4. none of these

Show answer
Correct Option: B

If $p$ is false, $q$ is true, then which of the following is/are false?

maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

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

  2. $\sim p$

  3. $\sim p\Rightarrow q$

  4. $\sim q$

Show answer
Correct Option: A,D
Explanation:
 $p$  $q$  $p\Rightarrow q$ $\sim \left( p\Rightarrow q \right) $  $\sim p$  $\sim q$   $\sim p\Rightarrow q$
 F  T  T  F  T  F  T

Here we see that $\sim \left( p\Rightarrow q \right) $  and $\sim q$  are false

$p$: He is hard working.
$q$: He is intelligent.
Then $ \sim q\Rightarrow\sim p$, represents

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. If he is hard working, then he is not intelligent.

  2. If he is not hard working, then he is intelligent.

  3. If he is not intelligent, then he is not had working.

  4. If he is not intelligent, then he is hard working.

Show answer
Correct Option: C
Explanation:

p:she is hardworking
q:she is intelligent

~p:she is not hardworking
~q:she is not intelligent

~q=>~p means She is not intelligent implies she is not hardworking
Hence, Option C

$p:$ He is hard working.
$q:$ He will win.
The symbolic form of "If he will not win then he is not hard working", is

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $ p\Rightarrow q$

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

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

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

Show answer
Correct Option: C
Explanation:

Given $p:$ He is hard working

and $q:$ He will win
we get $\sim p:$ He is not hard working

and $\sim q:$ He will not win
Now the given statement in the question is "If he will not win then he is not hard working" which means 
"If he will not win then he is not hard working"
For this conditional statement, the symbolic form is $\left( \sim q \right) \Rightarrow \left( \sim p \right) $

Simplify $(p\vee q)\wedge(p\vee\sim q)$

maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $p$

  2. $\sim p$

  3. $\sim q$

  4. $q$

Show answer
Correct Option: A
Explanation:

$(p\vee q)\wedge (p\vee \sim q)$
$=p\vee(q\wedge \sim q)$ (distributive law)
$=p\vee 0$ (complement law)
$=p$ ($0$ is indentity for v)

The negation of the statement "No slow learners attend this school," is:

maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. All slow learners attend this school.

  2. All slow learners do not attend this school.

  3. Some slow learners attend this school.

  4. Some slow learners do not attend this school.

  5. No slow learners do not attend this school.

Show answer
Correct Option: C
Explanation:

The negation is : It is false that no slow learners attend this school. Therefore, some slow learners attend this school.

Dual of $( p \rightarrow q ) \rightarrow r$ is _________________.

business maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

  1. $p\vee (\sim  q\wedge r)$

  2. $p\vee q\wedge r$

  3. $p\vee (\sim  q\wedge \sim r)$

  4. $\sim p\vee (\sim q\wedge r)$

Show answer
Correct Option: A