Question

Prove that √3 is irrational. You may use the fact that n2 is divisible by 3 only if n is divisible by 3.

Prove by contradiction that there is not a smallest positive rational number.

Answer #1

Let

Assume that there exists some integers such that a and b have no common factors. This means the fraction is in lowest terms.

Squaring on both sides, we get

-------(1)

(1) shows that a^2 is divisible by 3 and so a is divisible by 3

Therefore, a = 3p ---(A)

where p is some integer

Now substitute a = 3p in (1), we get

From this it can be said that b^2 is divisible by 3 and so b is divisible by 3

So, b = 3q ---(B)

From (A) and (B) we see that a and b have common factor 3 which is a contradiction.

Hence we cannot write

So, √3 is irrational

Prove that (17)^(1/3) is irrational. You may use the fact that
if n^3 is divisible by 17 then n is divisible by 17

10. (a) Prove by contradiction that the sum of an irrational
number and a rational number must be irrational. (b) Prove that if
x is irrational, then −x is irrational. (c) Disprove: The sum of
any two positive irrational numbers is irrational

Prove that there is no positive integer n so that 25 < n2
< 36. Prove this by directly proving the negation. Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides, or by multiplying both sides by a positive number. You
cannot use the square root function. Do not write a proof by
contradiction.

If you are given the fact that if m, n ∈ Z such that mn is
divisible by a prime number p, then m or n is divisible by p. How
do you use this to prove that for all n ∈ Z, √ n is rational if and
only if there exists m ∈ Z such that n = m^2?

Prove by contradiction that 5√ 2 is an irrational number. (Hint:
Dividing a rational number by another rational number yields a
rational number.)

: Prove by contradiction that 5√ 2 is an irrational number.
(Hint: Dividing a rational number by another rational number yields
a rational number.)

Irrational Numbers
(a) Prove that for every rational number µ > 0, there exists
an irrational number λ > 0 satisfying λ < µ.
(b) Prove that between every two distinct rational numbers there
is at least one irrational number. (Hint: You may find (a)
useful)

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

In the style of the proof that square root of 2 is irrational,
prove that the square root of 3 is irrational. Remember, we used a
proof by contradiction. You may use the result of Part 1 as a
"Lemma" in your proof.

Ex 2. Prove by contradiction the following claims. In each proof
highlight what is the contradiction (i.e. identify the proposition
Q such that you have Q ∧ (∼Q)).
Claim 1: The sum of a rational number and an irrational number
is irrational. (Recall that x is said to be a rational number if
there exist integers a and b, with b 6= 0 such that x = a b ).
Claim 2: There is no smallest rational number strictly greater...

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