Question

Prove that for all real numbers x, if x 2 is irrational, then x is irrational.

Answer #1

Prove the statement " For all real numbers r, if r is
irrational, then r/2 is irrational ". You may use any method you
wish. Be sure to state what method of proof you are using.

Prove the statement
For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ =
2⌊x⌋.

Let x ∈ ℝ. Prove that if x is irrational, then 2 + x is
irrational

1) Prove that for all real numbers x and y, if x < y, then x
< (x+y)/2 < y
2) Let a, b ∈ R. Prove that:
a) (Triangle inequality) |a + b| ≤ |a| + |b| (HINT: Use Exercise
2.1.12b and
Proposition 2.1.12, or a proof by cases.)

Define f: R (all positive real numbers) -> R (all positive
real numbers)
by f(x)= sqrt(x^3+2)
prove that f is bijective

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)

Prove the following using the specified technique:
(a) Prove by contrapositive that for any two real numbers,x and
y,if x is rational and y is irrational then x+y is also
irrational.
(b) Prove by contradiction that for any positive two real
numbers,x and y,if x·y≥100 then either x≥10 or y≥10.
Please write nicely or type.

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 between any two rational numbers there is an
irrational number.

Prove, that between any rational numbers there exists
an irrational number.

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