Let A = R x R, and let a relation S be defined as: “(x1 ,...
Let A = R x R, and let a relation S be defined as: “(x1 , y1 ) S
(x2 , y2 ) ⬄ points (x1 , y1 ) and (x2 , y2 ) are 5 units apart.”
Determine whether S is reflexive, symmetric, or transitive. If the
answer is “yes,” give a justification (full proof is not needed);
if the answer is “no” you must give a counterexample
Let A be the set of all lines in the plane. Let the relation R
be...
Let A be the set of all lines in the plane. Let the relation R
be defined as:
“l1 R l2 ⬄ l1 intersects
l2.” Determine whether S is reflexive, symmetric, or
transitive. If the answer is “yes,” give a justification (full
proof is not needed); if the answer is “no” you must give a
counterexample.
Let F be an ordered field. Let S be the subset [a,b)
i.e, {x|a<=x<b, x element of...
Let F be an ordered field. Let S be the subset [a,b)
i.e, {x|a<=x<b, x element of F}. Prove that infimum and
supremum exist or do not exist.
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C...
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C = {u, v, w},
Define f : A→B by f(p) = m, f(q) = k, f(r) = l, and f(s) = n, and
define g : B→C by g(k) = v, g(l) = w, g(m) = u, and g(n) = w. Also
define h : A→C by h = g ◦ f. (a) Write out the values of h. (b) Why
is it that...
Let S denote the set of all possible finite binary strings, i.e.
strings of finite length...
Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...
Let X = { a, b, c } and consider the ralation R on X given...
Let X = { a, b, c } and consider the ralation R on X given by R
= {(a,a),(b, b),(c, c),(a,b),(b,c),(a, c),(c,a)}
Is R symmetric? Explain
Is R transitive? Explain
Is R reflexive? Explain
Remeber to explain your answer. Thanks.
. Write down a careful proof of the following. Theorem. Let (a,
b) be a possibly...
. Write down a careful proof of the following. Theorem. Let (a,
b) be a possibly infinite open interval and let u ∈ (a, b). Suppose
that f : (a, b) −→ R is a function and that for every sequence an
−→ u with an ∈ (a, b), we have that lim f(an) = L ∈ R. Prove that
lim x−→u f(x) = L.