Question

2.

**Let A, B, C be subsets of a universe U.**

**Let R ⊆ A × A and S ⊆ A × A be binary relations on
A.**

i. If R is transitive, then R−1 is transitive.

ii. If R is reflexive or S is reflexive, then R ∪ S is reflexive.

iii. If R is a function, then S ◦ R is a function.

iv. If S ◦ R is a function, then R is a function

Answer #1

1. Let A, B, C be subsets of a universe U.
i. If A is contained in B, then C \ A is contained in C \ B.
ii. If A and B are disjoint, then A \ C and B \ C are
disjoint.
iii. If A and B are disjoint, then A ∪ C and B ∪ C are
disjoint.
iv. If A ⊆ (B ∪ C), then A ⊆ B or A ⊆ C.

Let A and B be two subsets of a universe U where |U| = 120.
Suppose that
|A^c ∩ B^c| = 25 and |A − B| = 15. Furthermore, there is a
bijection f : A → B. Find
|A ∩ B|. Show all the steps involved in obtaining the answer,
providing an explanation
for each step.

Determine whether the binary relation R on {a, b,
c} where R={(a, a), (b, b)), (c, c), (a, b), (a,
c), (c, b) } is:
a.
reflexive, antisymmetric, symmetric
b.
transitive, symmetric, antisymmetric
c.
antisymmetric, reflexive, transitive
d.
symmetric, reflexive, transitive

6. Let S be a finite set and let P(S) denote the set of all
subsets of S. Define a relation on P(S) by declaring that two
subsets A and B are related if A ⊆ B.
(a) Is this relation reflexive? Explain your
reasoning.
(b) Is this relation symmetric? Explain your
reasoning.
(c) Is this relation transitive? Explain your
reasoning.

Let X = { a, b, c } and consider the ralation R on X given by R
= {(a,a),(b, b),(c, c),(a,b),(b,c),(a, c),(c,a)}
Is R symmetric? Explain
Is R transitive? Explain
Is R reflexive? Explain
Remeber to explain your answer. Thanks.

Let X be a set and A a σ-algebra of subsets of X.
(a) A function f : X → R is measurable if the set {x ∈ X : f(x)
> λ} belongs to A for every real number λ. Show that this holds
if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ
∈ R. (b) Let f : X → R be a function.
(i) Show that if...

Let A, B be non-empty subsets of R. Define A + B = {a + b | a ∈
A and b ∈ B}. (a) If A = (−1, 2] and B = [1, 4], what is A + B?

Let
U={1,2, 3, ...,3200}.
Let S be the subset of the numbers in U that are multiples of
4, and let T be the subset of U that are multiples of 9. Since
3200 divided by 4 equals it follows that
n(S)=n({4*1,4*2,...,4*800})=800
(a) Find n(T) using a method similar to the one that showed
that n(S)=800
(b) Find n(S∩T).
(c) Label the number of elements in each region of a two-loop
Venn diagram with the universe U and subsets S...

2. Let R be a relation on the set of integers ℤ defined by ? =
{(?, ?): a2 + ?2 ?? ? ??????? ??????} Is this
relation reflexive? Symmetric? transitive?

Let A be the set of all lines in the plane. Let the relation R
be defined as:
“l1 R l2 ⬄ l1 intersects
l2.” Determine whether S is reflexive, symmetric, or
transitive. If the answer is “yes,” give a justification (full
proof is not needed); if the answer is “no” you must give a
counterexample.

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