Question

Prove the following: Theorem. Let R ⊆ X × Y and S ⊆ Y × Z be relations. Then

1. Range(S ◦ R) ⊆ Range(S), and

2. if Domain(S) ⊆ Range(R), then Range(S ◦ R) = Range(S)

Answer #1

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a) Explain why R is not an
equivalence relation. b) Explain why S is not an equivalence
relation. c) Find S ◦ R. d) Show that S ◦ R is an equivalence
relation. e) What are the equivalence classes of S ◦ R?

Prove the following theorem:
Theorem. Let a ∈ R
and let f be a function defined on an
interval centred at a.
IF f is continuous at a
and f(a) > 0 THEN
f is strictly positive on some interval
centred at a.

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

Let T: R^3----> R^3 where T(x,y,z) = (x-2z,y+z,x+2y) . Is T a
one-to-one transformation?
Is the range of T R^3 ? Explain

(a) Let f(z) = z^2. R is bounded by y = x, y= -x and x = 1. Find
the image of R under the mapping f.
(b)Find the all values of (-i)^(i)
(c)Find the all values of (1-i)^(4i)

Let R be an integral domain. Prove that R[x] is an integral
domain.

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

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