Question

Use proof by contradiction to prove the statement given. If a
and b are real numbers and 1 < a < b, then
a^{-1}>b^{-1}.

Answer #1

N.B: In the question it will be a-1<b-1

Use proof by contradiction to prove that if T is a tree, then
every edge of T is a bridge.

Proof by contradiction: Suppose a right triangle has side
lengths a, b, c that are natural numbers. Prove that at least one
of a, b, or c must be even. (Hint: Use Pythagorean Theorem)

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.

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 by contradiction: Let a and b be integers. Show that if is
odd, then a is odd and b is odd.
a) State the negation of the above implication.
b) Disprove the negation and complete your proof.

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.)

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

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.

Prove the following using Field Axioms of Real
Numbers. prove (b^(−1))^−1=b

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

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