DIY Questions Proofs Workshop

1 Prove that x +

1 ≥ 2 for all x > 0, x

a) directly b) by contradiction

2 Prove that (1+ a + a 2 +…+ a n!1 ) =

a n !1 , where a and n are positive integers, and a ≠ 1. a !1

a) by induction on n b) by setting S = 1+ a + a 2 +…+ a n!1 and subtracting it from aS. 3 Prove that n2 is odd if and only if n is odd. 4 Draw a conclusion from the following sentences, using all the information given: a)

Everyone who is sane can do logic;

b)

No insane person is fit to serve on a jury;

c)

None of your sons can do logic.

5 a) If the difference of two integers a and b is an even integer, prove that the product ab is the difference of two squares. b) State the converse of this theorem. Is the converse true? 6 Use a contrapositive argument to prove that: if a2 + b2 = c2, then either a or b is divisible by 3, where a, b and c are positive integers. Hint: what can the remainder be when you divide the square of an integer by 3?

7

⎛ 1 1 2 ⎞ ⎜ ⎟ Let A = ⎜ 0 1 1 ⎟ . ⎜ 0 0 1 ⎟ ⎝ ⎠

8

a) Can you find integers a, b such that a 2 + b2 = 47 ?

⎛ 1 n ⎜ n Prove by induction, that for all positive integers n, A = ⎜ 0 1 ⎜⎜ ⎝ 0 0

1 2

( n 2 + 3n) ⎞ ⎟ ⎟ n ⎟⎟ 1 ⎠

b) 47 = 4×11 + 3. Extend the result in a) to any number of the form 4k + 3, k ∈ℕ? 9

Let {an} be a sequence of integers satisfying: a1 = 2, a2 = 8 and an+1 = 4(an − an 1) for n ≥ 2. Guess a formula for an , and prove it using strong induction. (Hint: try looking at an/n) −

10

Prove the following theorem, using a contrapositive argument: If a and b are real numbers such that the product ab is irrational, then either a or b is irrational.

11

Prove that every fourth Fibonnaci number is divisible by 3, i.e. f4n is a multiple of 3, where f1 = f2 = 1, and fn = fn 1 + fn 2 for n ≥ 3. [The first few terms are: 1, 1, 2, 3, 5, … ] Hint: try induction −

12

−

Form a conjecture concerning the integer 3n − 1 for positive integers n and prove your conjecture.

Shirleen Stibbe

http://www.shirleenstibbe.co.uk

13

Is it possible to choose 55 different numbers between 1 and 100, so that no two of the numbers differs by a) 10? b) 11? You may find the grids below useful

14

Prove that nk is even if and only if n is even, where n is an integer, and k is a positive integer. Hint: You may find the binomial theorem useful in answering this question:

(s +1) k = s k +

( k1 ) s + ( k2 ) s k!1

k!2

+…+

( k k!1 ) s +1

where

( kr ) = r!(kk!! r)!

is an integer

15 Prove that x2 < x if and only if 0 < x < 1, for x ∈ ℝ 16 (a)

Give a direct proof to show that if 2n + 1 = m2 for the positive integers m and n, then {m, n, n +1} is a Pythagorean triple.

(b)

How does this result show that any odd positive integer greater than 1 can be one of the integers in a Pythagorean triple?

(c)

What does the result of Question 5 tell us about any Pythagorean triple?

Note: The ordered set of positive integers {a, b, c} is said to be a Pythagorean triple if and only if c2 = a2 + b2.

a

c b

17

Prove that if p is a prime number, and p2 + 2 is also prime, then p = 3.

18

Prove that n2 – 1 is divisible by 8 if and only if n is odd, where n is an integer.

19

Prove by induction that (You may assume

20

d n x = nx n!1 using the product rule, where n is a positive integer. dx

( )

d ( x) = 1 .) dx

Give a direct proof to show that if a and b are any two positive real numbers, then a + b ≥ 2√(ab).

21

Let n, a, b be positive integers. If a and b are factors of n, and ab < n, then ab is a factor of n. Prove or disprove.

Shirleen Stibbe

http://www.shirleenstibbe.co.uk

Let S = {a1 , a2 ,!, a10 } be any set of ten integers.

22

By considering the last digit of each of the numbers:

n1 = a1 , n2 = a1 + a2 , n3 = a1 + a2 + a3 , ! n10 = a1 + a2 +!+ a10 , prove that

either or

a) one of the elements of S is a multiple of 10 b) the sum of two or more elements of S must be a multiple of 10

Can you generalise this for any set of m integers (i.e. replace 10 by m throughout, and consider the numbers modulo m instead of the last digit of the numbers)? Hint: The Pigeonhole Principle says: Given n boxes and m > n objects, at least one box must contain more than one object. It was first stated by Dirichlet in 1834. 23

Write out truth tables for a) [(p ∨ q) ∧ (p ∨ r)] and b) [p ∨ (q ∧ r)] Hence show they are logically equivalent. You may find the following table a helpful starting point: p

q

r

T

T

T

T

T

F

T

F

T

T

F

F

F

T

T

F

T

F

F

F

T

F

F

F

(p ∨ q)

(p ∨ r)

(p ∨ q) ∧ (p ∨ r)

(q ∧ r)

24

Prove that 2(x2 + y2) ≥ (x + y)2 for all x, y ∈ ℝ.

26

Write out the truth tables for the following propositions, and hence show that they are equivalent: a)

¬ (p ⇒ q)

b)

p ∧ ¬q

p ∨ (q ∧ r)

[ not-( p implies q) ] or [ (p implies q) is false ]

[ p and not-q] or [ p is true and q is false ]

27

f n be the nth Fibonacci number, where f1 = 1, f 2 = 1 and f n = f n−1 + f n−2 for n ≥ 3. Prove that f 2 + f 4 + … + f 2 n = f 2 n+1 − 1 for all positive integers n.

28

Let S = {x ∈ ℝ: x2 < x} and T = {x ∈ ℝ: 0 < x < 1}.

Let

Prove that S = T. 29

Prove by induction that 4n + 2 is divisible by 3 for all integers n ≥ 0.

30

Prove that the difference of two integers a and b is an even integer if and only if a and b have the same parity (ie a and b are both even or both odd).

32

Prove by induction that, for all integers n ≥ 2

(1! a1 )(1! a2 ) … (1! an ) > 1! (a1 + a2 +… an ) where 0 < ak < 1 for all k " ℝ

Shirleen Stibbe

http://www.shirleenstibbe.co.uk

31

Let S1 and S2 be finite subsets of a vector space, where S1 is a subset of S2. Show that: a) If S1 is linearly dependent, then so is S2. b) If S2 is linearly independent, then so is S1. Reminder: A finite set of vectors {v1 , v2 , ... , vk } is linearly dependent of there exist real numbers, c1 , c2 ,…, ck , not all zero, such that c1v1 + c2 v2 + ... + ck vk = 0 A finite set of vectors is linearly independent if it is not linearly dependent.

33

Prove or disprove: a) If a < b, c < d and a < c, then b < d . b)

If a > b > 0 and c > 0 then

a a+c > >1 b b+c

You may find the following rules helpful: i) ( p < q) ! ( p + r < q + r), p, q, r " ℝ ii) If r > 0, then (p < q) ! ( pr < qr), p, q " ℝ, r ∈ ℝ+ 34

Is there a smallest real number strictly greater than

35

Let n be a positive integer. The sets A1 , A2 ,…, An are closed intervals on the real number line, where

1

2

? Justify your answer.

Ar = [r, r + n] . What is A1 ! A2 !…! An ? Justify your answer.

Shirleen Stibbe

http://www.shirleenstibbe.co.uk