Question

Prove the following: (By contradiction)

If p,q are rational numbers, with p<q, then there exists a rational number x with p<x<q.

Answer #1

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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

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...

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

Prove that the only ring homomorphism from Q (rational numbers)
to Q (rational numbers) is the identity map.

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)

6. Define S={q∈Q:−√3< q <√3}, where Q denotes the rational
numbers. Prove that S can not be the set of subsequential limits of
any sequence (sn) of reals.
Please be very verbose. I already have the answer to this
question as a contradiction. It is okay if you use a contradiction,
however please explain thoroughly.

Recall that Q+ denotes the set of positive rational numbers.
Prove that Q+ x Q+ (Q+ cross Q+) is countably infinite.

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.)

If p is a rational # and q is an irrrational #
Is the number p/q rational or irrational and prove your
answer.

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