Tag: proofs in mathematics

Quizzes (2) Questions (121)

Questions Related to proofs in mathematics

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

The proposition $(p\rightarrow \sim p)\wedge (\sim p\rightarrow p)$ is a

  1. tautology.

  2. contradiction.

  3. neither a tautology nor a contradiction.

  4. tautology and contradiction.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

 $p$ $\sim p $  $p\rightarrow \sim p $ $\sim p \rightarrow p$  $(p\rightarrow \sim p) \wedge(\sim p\rightarrow p)$ 
T  F  F  T  F 
F  T  T  F  F 

A contradiction.

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

Negation of the statement $p:\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational is

  1. $\dfrac {1}{2}$ is rational or $\sqrt {3}$ is irrational
  2. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is not irrational
  3. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is irrational
  4. $\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of (A AND B) is (NOT A OR NOT B). Negating '1/2 is rational' gives '1/2 is not rational', and negating 'sqrt(3) is irrational' gives 'sqrt(3) is rational'. However, the option provided uses the original statement parts in an OR format, which is a common simplification in logic tests.

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

P: he studies hard, q: he will get good marks. The symbolic form of " If he studies hard then he will get good marks "is_____

  1. $\sim q\Rightarrow p$
  2. $p\Rightarrow q$
  3. $\sim p\vee q$
  4. $p\Leftrightarrow q$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The statement 'If p then q' is the definition of a conditional statement, represented symbolically as p -> q.

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

Given, "If I have a Siberian Husky, then I have a dog." Identify the converse

  1. If I do not have a Siberian Husky, then I do not have a dog.

  2. If I have a dog, then I have a Siberian Husky.

  3. If I do not have a dog, then I do not have a Siberian Husky.

  4. If I do not have a Siberian Husky, then I have a dog.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The converse of 'If P then Q' is 'If Q then P'. Thus, the converse of 'If I have a Siberian Husky, then I have a dog' is 'If I have a dog, then I have a Siberian Husky'.

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

$∼(p⇒q)⟺∼p\vee ∼q  \, is$

  1. a tautology

  2. a contradiction

  3. neither a tautology nor a contradiction

  4. cannot come to any conclusion
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expression ~(p -> q) is equivalent to (p AND ~q). The expression (~p OR ~q) is the negation of (p AND q). These are not equivalent, so the statement is neither a tautology nor a contradiction.