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.
On this page, you should learn to
Make simple algebraic deductive proofs Use the symbol \(\equiv\) for identities Here is a quiz that practises the skills from this page
START QUIZ! Complete the following to "prove that 2 consecutive integers add together to make an odd integer"
n 2n + 1 2n n + 1
\(n + (n + 1) \equiv2n+1\)
2n is an even number, so 2n + 1 is odd
Check
Work out a and b to prove that x² - 4x + 7 > 0
We can complete the square:
x² - 4x + 7 \(\equiv\) (x - a )² + b
A square number is always greater than or equal to zero, so (x - a )² ≥ 0
Since, b ≥ 0,
(x - a )² + b ≥ 0
Hence x² - 4x + 7 > 0
This is a good technique for showing that a function is not negative
Check
Complete the following to "prove that adding 2 consecutive odd numbers gives a multiple of 4"
2n 4n 2n + 1 4n - 1
4n is divisible by 4, hence is a multiple of 4
Check
Complete the following to "prove that if m and n are either both even or both odd, then m + n is even"
k 2k - 1 j + k j + k - 1 2k
Check
Complete the following to "prove that one more than the product of two consecutive odd integers is the square of an even number"
2n² 2n 2n + 1 4n² 2n - 1
Finding the product means that we multiple the two numbers together
Check
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 b) Start with the left hand side. Expand brackets and simplify Full Solution
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 b) Start with the left hand side, add the algebraic fractions together by making the denominators the same. Full Solution
Prove that the sum of three consecutive integers is divisible by 3
Hint Let the first integer be n
What would be the value of the next integer?
Full Solution
a) Verify that x² - 4x + 5 is positive for x = -1
b) Prove that x² - 4x + 5 is positive for all x
Hint A square number cannot be negative. Is it posible to write this expression as a square? Full Solution
Prove that the difference between the square of any two consecutive odd integers is divisible by 8
Hint An odd number can be written as 2n - 1.
A number is divisible by 8 if it has a factor of 8
Full Solution
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 \(^nC_r=\frac{n!}{(n-r)!r!}\)
Some careful manipulation is required for this proof. Consider that \(r!=r\cdot(r-1)!\)
Full Solution MY PROGRESS
Self-assessment How much of Deductive Proofs have you understood?
My notes
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