Question

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these, list all ordered pairs in the relation

Answer #1

S={1,2,3}
a. Draw the directed graphs of all equivalence relations on
S
b. How many binary relations on S are symmetric? Justify

Let R be a relation on set RxR of ordered pairs of real numbers
such that (a,b)R(c,d) if a+d=b+c. Prove that R is an equivalence
relation and find equivalence class [(0,b)]R

Determine whether the given relation is an equivalence relation
on {1,2,3,4,5}. If the relation is an equivalence relation, list
the equivalence classes (x, y E {1, 2, 3, 4, 5}.)
{(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1), (3,4),
(4,3)}
If the relation above is not an equivalence relation, state that
the relation is not an equivalence relation and why.
Example: "Not an equivalence relation. Relation is not
symmetric"
Remember to test all pairs in relation R

Let R1 and R2 be equivalence relations on a set A. (a) Must
R1∪R2 be an equivalence relation? (b) Must R1∩R2 be an equivalence
relation? (c) Must R1⊕R2 be an equivalence relation?[⊕is the
symmetric difference:x∈A⊕B if and only if x∈A,x∈B, and x
/∈A∩B.]

There is no equivalence relation R on set {a, b, c, d,
e} such that R contains less than 5 ordered pairs (True or
False)

Let S1 and S2 be any two equivalence relations on some set A,
where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.
Prove or disprove (all three):
The relation S defined by S=S1∪S2 is
(a) reflexive
(b) symmetric
(c) transitive

Let S1 and S2 be any two equivalence relations on some set A,
where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.
Prove or disprove (all three):
The relation S defined by S=S1∪S2 is
(a) reflexive
(b) symmetric
(c) transitive

Determine the distance equivalence classes for the relation R is
defined on ℤ by a R b if |a - 2| = |b - 2|.
I had to prove it was an equivalence relation as well, but that
part was not hard. Just want to know if the logic and presentation
is sound for the last part:
8.48) A relation R is defined on ℤ by a R b if |a - 2| = |b -
2|. Prove that R...

List all equivalence relations on {0, 1, 2,
3}.

Let R be an equivalence relation defined on some set A.
Prove using mathematical induction that R^n is also an
equivalence relation.

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