Deductive Proofs

On this page, we will look at deductive reasoning in order to be able to make direct proofs. This is a hugely important topic in mathematics, since we like to be absolutely sure of the results we have found. However, this can be a challenging - when a problem is unfamiliar, it is often difficult to know where to start. The key is to be able to use our existing knowledge and theorems, make connections and work towards a conclusion. You might want to think of these as puzzles to solve.


Key Concepts

On this page, you should learn to

  • Make simple algebraic deductive proofs
  • Use the symbol \(\equiv\) for identities

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

a) Verify that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2=4x^3 \) for x = 3

b) Prove that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2\equiv4x^3 \) for all x

Hint

Full Solution

 

Question 2

a) Verify that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=\frac{2x+1}{x^2\left(x+1\right)^2} \) for x = -2

b) Prove that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\equiv\frac{2x+1}{x^2\left(x+1\right)^2} \) for all x

Hint

Full Solution

 

Question 3

Prove that the sum of three consecutive integers is divisible by 3

Hint

Full Solution

Question 4

a) Verify that x² - 4x + 5 is positive for x = -1

b) Prove that x² - 4x + 5 is positive for all x

Hint

Full Solution

 

Question 5

Prove that the difference between the square of any two consecutive odd integers is divisible by 8

Hint

Full Solution

 

Question 6

a) Verify that \(^3 C_1\ +\ ^3 C_2\ =\ ^4C_2\)

b) Prove that \(^{n-1} C_{r-1}\ +\ ^{n-1} C_r\ =\ ^nC_r\)

Hint

Full Solution

MY PROGRESS

How much of Deductive Proofs have you understood?