Question

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

Answer #1

1. Prove that the sum of any rational number with an irrational
number must be irrational.
2. Prove or disprove: If a,b, and c are integers such that
a|(bc), then a|b or a|c.

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)

Prove that between any two rational numbers there is an
irrational number.

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

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...

(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

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.

Prove that if p is a positive rational number, then √p + √2 is
irrational.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 10 minutes ago

asked 40 minutes ago

asked 59 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago