Question

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.

Answer #1

1.

Disprove: The following relation R on set Q is either reflexive,
symmetric, or transitive. Let t and z be elements of Q. then t R z
if and only if t = (z+1) * n for some integer n.

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?

Consider the relation R defined on the real line R, and defined
as follows: x ∼ y if and only if the distance from the point x to
the point y is less than 3. Study if this relation is reflexive,
symmetric, and transitive. Which points are related to 2?

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...

Problem 3
For two relations R1 and
R2 on a set A, we define the
composition of R2 after R1
as
R2°R1 = { (x,
z) ∈ A×A | (∃ y)( (x,
y) ∈ R1 ∧ (y, z) ∈
R2 )}
Recall that the inverse of a relation R, denoted
R -1, on a set A is defined as:
R -1 = { (x, y) ∈
A×A | (y, x) ∈ R)}
Suppose R = { (1, 1), (1, 2),...

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 }.

Let us say that two integers are near to one another provided
their difference is 2 or smaller (i.e., the numbers are at most 2
apart). For example, 3 is near to 5, 10 is near to 9, but 4 is not
near to 8. Let R stand for this is-near-to relation. (a) Write down
R as a set of ordered pairs. Your answer should look like this: R =
{(x, y) : . . .}. (b) Prove or disprove:...

4. Prove that {(x, y) ∈ R 2 ∶ x − y ∈ Q} is an equivalence
relation on the set of real numbers, where Q denotes the set of
rational numbers.

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,...

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

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