Question

Determine whether or not the following relations have the properties of the relations. Be sure to justify your answers.

a) R = {(x, y)| y is a biological parent of x} on the set of all people

b) R = {(x, y) ∈ N × N | lcm(x, y) = 10}

Answer #1

A relation is said to be relation in mathematical terms, if it is well defined. In other words, one should be able to decide without any ambiguity that two elements are related or not.

For example, if we consider the cities nearby to Delhi, then this relation is not well defined, because the word "nearby" is not well defined itself.

On the other hand, if we consider the the cities which are within 100 Km of Delhi, the this represents a relation because the distance 100 Km is well defined.

a) R = {(x, y)| y is a biological parent of x} on the set of all people

Here R is a relation because it is well defined that a person is biological parent of other or not. There is no ambiguity in statement.

b) R = {(x, y) ∈ N × N | lcm(x, y) = 10}

Here R is a relation because the it is can be decided without ambiguity that lcm of two given numbers is 10 or not.

For each of the following relations, determine whether the
relation is reﬂexive, irreﬂexive, symmetric, antisymmetric, and/or
transitive. Then ﬁnd R−1.
a) R = {(x,y) : x,y ∈Z,x−y = 1}.
b) R = {(x,y) : x,y ∈N,x|y}.

Determine whether the relations R1 and R2 are equivalence
relations to the specified quantity and, if necessary, determine
the corresponding equivalence classes. ∀x, y ∈Z: x ~R1 y ⇐⇒ x + y
is divisible by 2, ∀ g, h ∈ {t: t is a straight line in R2}: g ~R2
h ⇐⇒ g and h have common Points.

Determine whether the relations R1 and R2 are equivalence
relations to the specified quantity and, if necessary, determine
the corresponding equivalence classes.
∀x, y ∈Z: x ~R1 y ⇐⇒ x + y is divisible by 2,
∀ g, h ∈ {t: t is a straight line in R2}: g ~R2 h ⇐⇒ g and h
have common Points.

For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning). For countable
sets, provide some explicit counting scheme and list the first 20
elements according to your scheme. (a) The set [0, 1]R ×
[0, 1]R = {(x, y) | x, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
(b) The set [0, 1]Q × [0, 1]Q = {(x, y) |
x, y ∈ Q, 0 ≤ x ≤...

For each of the following relations on the set of all integers,
determine whether the relation is reflexive, symmetric, and/or
transitive:
(?, ?) ∈ ? if and only if ? < ?.
(?, ?) ∈ ? if and only ?? ≥ 1.
(?, ?) ∈ ? if and only ? = −?.
(?, ?) ∈ ? if and only ? = |?|.

let A = {−4, 4, 5, 8} and B = {4, 5, 6} and define relations R
and S from A to B as follows:
For all elements (x in A , y in B) , x R y ⇔ |x| = |y| + 1 and x
S y ⇔ x /y is an integer.
1. Find A X B and A intersect B.
2. Is the relation R reflexive ? Justify your answer.

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 }.

For each of the properties reflexive, symmetric, antisymmetric,
and transitive, carry out the following.
Assume that R and S are nonempty relations on a set A that both
have the property. For each of Rc, R∪S, R∩S, and R−1, determine
whether the new relation
must also have that property;
might have that property, but might not; or
cannot have that property.
A ny time you answer Statement i or Statement iii, outline a
proof. Any time you answer Statement ii,...

Determine and explain the following properties of convolution
for y[n] = x[n] ∗ h[n]
a) If x[n] is an even function and h[n] is an even function, is
y[n] even,odd, or neither?
b) If x[n] is an odd function and h[n] is an odd function, is
y[n] even,odd, or neither?
c) If x[n] is an even function and h[n] is an odd function, is
y[n] even,odd, or neither?

1.28 Dectermine which of the properties listed in 1.27 hold and
which do not hold for each of the following discrite-time systems.
Justify your answers. In each example, y[n] denotes the system
output and x[n] is the system input.
b) y[n] = x[n-2]-2x[n-8]
c) y[n]=nx[n]
system may or may not be:
1) Memoryless
2) Time invariant
3) Linear
4) causal

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