Exercise.
(a) Find the set of all
-3,-5
such that
+1
+3
+3
+5
.
(b) Find the set of all
-3,-4
such that
+1
+3
+3
+4
.
(c) Find the set of all
-3,-2
such that
+1
+3
+3
+2
.
Give your answers as an interval or a union of disjoint intervals.
Exercise.
(a) Find the set of all
such that
2-1
.
(b) Find the set of all
such that 3
2+1
.
Give your answers as an interval or a union of disjoint intervals.
Leave any numbers in your answers in surd form.
Exercise.
Consider the sequence defined by
=
+1
.
Prove that
1
as follows.
(a) Find as simple an expression as you can for
-1
.
Make sure your expression does not involve absolute value signs.
(b) Supposing
0
is given, use your answer to (a) to find
the set of
such that
0
-1
writing the answer as
()
, where
is a finite set of naturals numbers and ()
is
an expression involving only.
(Hint: By conjugating surds, show that
-1
12
for all . You don't have to say what your set
is. Why is it finite? Be sure to use
not . If you have
()
you can replace it with
()+1
if you change
the set .)
(c) Write down a proof that
1
starting with
Let
0
be arbitrary
and
Let =
()
.
Exercise.
Consider the statement X, which is
0
0
-
-
and the sequence defined by (
)=-1
+1
.
Prove that X holds for this sequence for both =1 and
=-1.
All assertions you make in your answers MUST be supported by proofs.