Tag: proofs in mathematics

Quizzes (2) Questions (121)

Questions Related to proofs in mathematics

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

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

  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) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of an implication (p -> q) is (p AND NOT q). Applying this to (~p -> q), we get (~p AND NOT q), which simplifies to (~p AND ~q).

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

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

  1. $q \vee r$
  2. $q \wedge r$
  3. $q \rightarrow r$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Applying Boolean algebra or logic laws to the expression (p and q) or (not p and q) or (not q and r), we can factor out q from the first two terms to get q, which then combines with the third term to simplify to q or r.

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

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

  1. $\sim (p\Rightarrow q)$
  2. $\sim p$
  3. $\sim p\Rightarrow q$
  4. $\sim q$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
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

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

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

  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.
Reveal answer Fill a bubble to check yourself
C Correct answer
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

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

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

  1. $ p\Rightarrow q$
  2. $ (\sim p)\Rightarrow (\sim q)$
  3. $ (\sim q)\Rightarrow (\sim p)$
  4. $ (\sim q)\Rightarrow p$
Reveal answer Fill a bubble to check yourself
C Correct answer
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) $

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

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

  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.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

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

To find the dual, replace AND with OR, OR with AND, True with False, and False with True. The dual of (p -> q) -> r is p OR (~q AND r).