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Methods of Mathematical Proof

Quiz covering various proof techniques including proof by contradiction, direct method, contrapositive, and counter examples

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

The contradiction of the statement to prove by contradiction method of the following will be
" If function is continuous then it is differentiable."

  1. Assume function is differentiable.
  2. Assume function is not continuous.
  3. Assume function is continuous.
  4. Assume function is not differentiable.
Question 2 Multiple Choice (Single Answer)

Which of the following is the correct steps to take when proving a statement using proof by contradiction?

  1. 1) Assume that your statement is true.

    2) Show this is the case using definitions and theorems.

    3) State that the statement is true.
  2. 1) Assume your statement is true for a certain instance.

    2) Show that it is true in more than one instance.

    3) State that your statement must be true.
  3. 1) Assume your statement to be false.

    2) Proceed as you would in a direct proof.

    3) Come across a contradiction.

    4) Use the contradiction to state that your assumption of the statement being false can't be the case, so your statement must be true.
  4. None of the answers are correct.
Question 3 Multiple Choice (Single Answer)

A pattern, plan, representation or description designed to show structure is known as a

  1. sample
  2. model
  3. design
  4. structure
Question 4 Multiple Choice (Single Answer)

The contrapositve of the statement If Mohan works hard, then he gets a first class is

  1. If Mohan gets first class, then he does not works hard
  2. If Mohan does not get a first class, then he works hard
  3. If Mohan does not get a first class, then he does not work hard
  4. If Mohan does not work hard, then he does not get a first class
Question 5 Multiple Choice (Single Answer)

To prove : "The integers can be of the form $4n,4n+1,4n+2  \ or \ 4n+3$" by direct method, we shall start the proof by the assumption 

  1. Let integers not of form $4n,4n+1,4n+2 \ or \ 4n+3$.
  2. Let integers be not of form $4n$.
  3. Let $z$ be any integer.
  4. Let $z<0$
Question 6 Multiple Choice (Single Answer)

To prove " If  $x,x\in N$  is even then $x^2$ is even". By direct method, we must start with the assumption:

  1. Let $x ^2 $ not even
  2. Let $x$ not even
  3. Let $x$ be even natural number.
  4. Let $x\notin N$
Question 7 Multiple Choice (Single Answer)

To prove: "The perpendicular from centre of a circle to the chord, bisects the chord." The proof started from assumption "Let OM be the perpendicular to chord AB". 

This method of proof is  

  1. The proof by contradiction
  2. The Direct method.
  3. Induction method.
  4. The proof by contrapositive method.
Question 8 Multiple Choice (Single Answer)

To prove: "If $f,g $ are continuous functions then $f+g$ is continuous." The proof started from assumption " Let $f,g$ be continuous functions."  

This method of proof is 

  1. The Direct method.
  2. The proof by contradiction.
  3. The proof by contrapositive method.
  4. Induction method.
Question 9 Multiple Choice (Single Answer)

If a triangle is equiangular, then it is an obtuse angled triangle. Which of the following statements doesn't convey the same meaning as of this mentioned sentence.

  1. A triangle is equiangular only if it is an obtuse angled triangle
  2. If a triangle is not obtuse angled triangle then it is not an equiangular triangle.
  3. Equiangularity is a sufficient condition for triangle to be obtuse angled.
  4. A triangle is only obtuse is obtuse angled if it is equiangular
Question 10 Multiple Choice (Single Answer)

To prove "" All prime numbers are not odd."  we showed that "$2$ is even and prime"
This method is  

  1. The Direct method.
  2. The proof by contradiction.
  3. Induction method.
  4. The proof by giving counter example.
Question 11 Multiple Choice (Single Answer)

To prove any preposition by "giving counter example" we must give at-least ______ example(s).

  1. One
  2. Two
  3. Three
  4. more than three
Question 12 Multiple Choice (Single Answer)

The proposition $(p , \Rightarrow , p),\wedge , (p , \Rightarrow  , p)$ is a

  1. Tautology
  2. neither tautology nor contradiction
  3. contradiction
  4. None of these
Question 13 Multiple Choice (Multiple Answers)

Which of the following is true for counter example.

  1. A counter example is an exception to a proposed general rule or law
  2. A counter example is a specific instance of the falsity of a universal quantification (a "for all" statement).
  3. Any hard-working student is a counter example to "all students are lazy"
  4. None of these
Question 14 Multiple Choice (Single Answer)

Counter example to the statement "All prime numbers are odd." is

  1. The prime number $2$
  2. The prime number $3$
  3. Number $1$
  4. None of these

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