Question

Consider an axiomatic system that consists of elements in a set S and a set P of pairings of elements (a, b) that satisfy the following axioms:

A1 If (a, b) is in P, then (b, a) is not in P.

A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in P.

Given two models of the system, answer the questions below.

M1: S= {1, 2, 3, 4}, P= {(1, 2), (2, 3), (1, 3)}

M2: Let S be the set of real numbers and let P

consist of all pairs (x, y) where x < y.

Q. Find another independent axiom A3 of the system, which is true in M1, but not in M2. Use this result to argue that the original system is not complete.

An example of such independent axiom:

A3 There is an element x in S such that (y, x) is not in P for any y in S. Then, A3 is true in M1 since there is 4 in S with no pairs in P containing. But, every element in S of M2, namely, every real number has infinitely many smaller numbers in S. Thus, M1 is a model of the augmented system containing A1, A2, A3, where as M2 is not. It concludes that the given system with A1 and A2 is not complete.

Answer #1

Using field and order axioms prove the following theorems:
(i) 0 is neither in P nor in - P
(ii) -(-A) = A (where A is a set, as defined in the axioms.
(iii) Suppose a and b are elements of R. Then a<=b if and
only if a<b or a=b
(iv) Let x and y be elements of R. Then either x <= y or y
<= x (or both).
The order axioms given are :
-A = (x...

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 <...

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

Define p to be the set of all pairs (l,m) in N×N such that l≤m.
Which of the conditions (a), (c), (r), (s), (t) does p satisfy?
(a) For any two elements y and z in X with (y,z)∈r and (z,y)∈r,
we have y=z
.(c) For any two elements y and z in X, we have (y,z)∈r or
(z,y)∈r.
(r) For each element x in X, we have (x,x)∈r.
(s) For any two elements y and z in X with...

Consider the set of all ordered pairs of real numbers with
standard vector addition but with scalar multiplication defined
by k(x,y)=(k^2x,k^2y).
I know this violates (alpha + beta)x = alphax + betax, but I'm
not for sure how to figure that out? How would I figure out which
axioms it violates?

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

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 and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

Let S be the set of all real circles, defined by (x-a)^2 +
(y-b)^2=r^2. Define d(C1,C2) =
√((a1 − a2)^2 + (b1 −
b2)^2 + (r1 − r2)^2 so that S is a
metric space.
Prove that metric space S is NOT a complete metric space. Give a
clear example.
Describe points of C\S as limits of the appropriate sequences of
circles, where C is the completion of S.

4. (Sec 2.5) Consider purchasing a system of audio components
consisting of a receiver, a pair of speakers, and a CD player. Let
A1 be the event that the receiver functions properly throughout the
warranty period, A2 the event that the speakers function properly
throughout the warranty period, and A3 the event that the CD player
functions properly throughout the warranty period. Suppose that
these events are (mutually) independent with P(A1) = .95, P(A2) =
.98 and P(A3) = .80....

1. Determine in each of the following cases, whether the
described system is or not a group. Explain your answers. Determine
what of them is an Abelian group.
a) G = {set of integers}, a* b = a − b
b) G = {set of matrices of size 2 × 2}, A * B = A · B
c) G = {a0, a1, a2, a3,
a4}, ai * aj = a|i+j|,
if i+j < 5, ai *aj = a|i+j−5|,
if...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 23 minutes ago

asked 49 minutes ago

asked 54 minutes ago

asked 54 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago