Question

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.]

Answer #1

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 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.

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

Let ~ be an equivalence relation on a given set A. Show [a] =
[b] if and only if a ~ b, for all a,b exists in A.

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

Consider these relations on the set of integers
R1 = { (a,b) | a < b or a ≥ b}
R2 = { (a,b) | a + b < 5 }
R3 = { (a,b) | a <= b }
R4 = { (a,b) | a = b +3 }
R5 = { (a,b) | a < b - 1 }
R6 = { (a,b) | a + 2 > b }
Choose following pairs that fit at least four...

1.1. Let R be the counterclockwise rotation by 90 degrees.
Vectors r1=[3,3] and r2=[−2,3] are not perpendicular. The inverse U
of the matrix M=[r1,r2] has columns perpendicular to r2 and r1, so
it must be of the form U=[x⋅R(r2),y⋅R(r1)]^T for some scalars x and
y. Find y^−1.
1.2. Vectors r1=[1,1] and r2=[−5,5] are perpendicular. The
inverse U of the matrix M=[r1,r2] has columns perpendicular to r2
and r1, so it must be of the form U=[x⋅r1,y⋅r2]^T for some scalars
x...

Prove/disprove the following claim: If R1 and R2 are integral
domains, then R1 ⊕ R2 must also be an integral domain under the
operations
• (r1,r2)+(s1,s2)=(r1 +s1,r2 +s2)
• (r1,r2)·(s1,s2)=(r1 ·s1,r2 ·s2)

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