Question

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, provide two examples: one where the new relation has the property, and onewhere the new relation does not. (You may use graphs to describe your examples.)

Answer #1

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.

Determine whether the following is reflexive, symmetric,
antisymmetric, transitive, and/or a partial order:
(x, y) ∈ R if 3 divides x – y

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}.

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
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(5) (12pts) Consider the < relation defined on R as usual, where
x <...

1. Suppose we have the following relation defined on Z. We say
that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼
defines an equivalence relation on Z. (b) Describe the equivalence
classes under ∼ .
2. Suppose we have the following relation defined on Z. We say
that a ' b iff 3 divides a + b. It is simple to show that that the
relation ' is symmetric, so we will leave...

Merck, AIDS, and Africa Written July 2001, Revised October 23,
2003 Merck was being pilloried in the international press. The
issue? Its role in AIDS crisis in Sub-Saharan Africa, where the
price of AIDS treatments far exceeded patients’ ability to pay. The
fallout from public opinion threatened not only Merck’s valued
reputation, but the international system of prices and intellectual
property rights on which Merck’s business was based. The
Pharmaceuticals Industry The pharmaceuticals industry is known for
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