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.

Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x, y) ∈ R if and only if x + 2 > y.
For example, (4, 3) is in R because 4 + 2 = 6, which is greater
than 3.
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.

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.

Let A = {1,2,3,4,5} and X = P(A) be its powerset. Define a
binary relation on X by for any sets S, T ∈ X, S∼T if and only if S
⊆ T.
(a) Is this relation reflexive?
(b) Is this relation symmetric or antisymmetric?
(c) Is this relation transitive?

Determine whether the relation R is reflexive, symmetric,
antisymmetric, and/or transitive [4 Marks]
22
The relation R on Z where (?, ?) ∈ ? if ? = ? .
The relation R on the set of all subsets of {1, 2, 3, 4} where
SRT means S C T.

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 }.

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.

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