Question

Prove the following statements by contradiction

a) If x∈Z is divisible by both even and odd integer, then x is even.

b) If A and B are disjoint sets, then A∪B = AΔB.

c) Let R be a relation on a set A. If R = R−1, then R is symmetric.

Answer #1

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

Consider the following relation on the set Z: xRy ?
x2 + y is even.
For each question below, if your answer is "yes", then prove it, if
your answer is "no", then show a counterexample.
(i) Is R reflexive?
(ii) Is R symmetric?
(iii) Is R antisymmetric?
(iv) Is R transitive?
(v) Is R an equivalence relation? If it is, then describe the
equivalence classes of R. How many equivalence classes are
there?

Perform the following tasks:
a. Prove directly that the product of an even and an odd number
is even.
b. Prove by contraposition for arbitrary x does not equal -2: if
x is irrational, then so is x/(x+2)
c. Disprove: If x is irrational and y is irrational, then x+y is
irrational.

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

1. Prove p∧q=q∧p
2. Prove[((∀x)P(x))∧((∀x)Q(x))]→[(∀x)(P(x)∧Q(x))]. Remember to
be strict in your treatment of quantifiers
.3. Prove R∪(S∩T) = (R∪S)∩(R∪T).
4.Consider the relation R={(x,y)∈R×R||x−y|≤1} on Z. Show that
this relation is reflexive and symmetric but not transitive.

Complete the following table. If a property does not hold give
an example to show why it does not hold.
If it does hold, prove or explain why. Use correct symbolism.
(Just Yes or No is incorrect)
R = {(a,b) | a,b ∃ Z: : a + b-even
S = {(a,b) | a,b ∃ Z: : a + b-odd
T = {(a,b) | a,b ∃ Z: : a + 2b-even
Relation
Reflexive
Symmetric
Anti Symmetric
Neither Symmetric or anti-symmetric
Transitive...

Determine whether the relation R on N is reﬂexive, symmetric,
and/or transitive. Prove your answer.
a)R = {(x,y) : x,y ∈N,2|x,2|y}.
b)R = {(x,y) : x,y ∈ A}. A = {1,2,3,4}
c)R = {(x,y) : x,y ∈N,x is even ,y is odd }.

Without using induction, prove that for x is an odd, positive
integer, 3x ≡−1 (mod 4). I'm not sure how to approach the problem.
I thought to assume that x=2a+1 and then show that 3^x +1 is
divisible by 4 and thus congruent to 3x=-1(mod4) but I'm stuck.

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