Question

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.

Answer #1

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

6. (a) Prove by contrapositive: If the product of two natural
numbers is greater than 100, then at least one of the numbers is
greater than 10. (b) Prove or disprove: If the product of two
rational numbers is greater than 100, then at least one of the
numbers is greater than 10.

Prove the following:
For any positive real numbers x and y, x+y ≥
√(xy)

Write the contrapositive statements to each of the following.
Then prove each of them by proving their respective
contrapositives.
a. If x and y are two integers whose product is even, then at
least one of the two must be even.
b. If x and y are two integers whose product is odd, then both
must be odd.

In each of the following, prove that the specified subset H is
not a subgroup of the given group G: (a) G = (Z, +), H is the set
of positive and negative odd integers, along with 0. (b) G = (R,
+), H is the set of real numbers whose square is a rational number.
(c) G = (Dn, ◦), H is the set of all reflections in G.

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

Prove the following statements using either direct or
contrapositive proof.
18. If a,b∈Z,then (a+b)^3 ≡ a^3+b^3 (mod 3).

Discreet Math: Prove or disprove each statement
a) For any real number x, the floor of 2x = 2 the floor of x
b) For any real number x, the floor of the ceiling of x = the
ceiling of x
c) For any real numbers x and y, the ceiling of x and the
ceiling of y = the ceiling of xy

Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers
in the interval (0, 1). (Hint: Prove this by contradiction.
Specifically, (i) assume that there are countably infinite real
numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii)
express each real number x1 between 0 and 1 in decimal expansion;
(iii) construct a number y whose digits are either 1 or 2. Can you
find a way...

Here are two statements about positive real numbers. Prove or
disprove each of the statements
∀x, ∃y with the property that xy < y2
∃x such that ∀y, xy < y2 .

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