Download 1. Prove the second part of De Morgan’s Laws, namely... A ∪ B = A ∩ B.

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Transcript
1. Prove the second part of De Morgan’s Laws, namely for sets A and B
A ∪ B = A ∩ B.
2. Show that one of the digits 1, 2, . . . , n occurs infinitely often in the decimal expansion of π
3. Decide whether the following sets are bounded by trying to find a upper/lower bound (note: it does
not need to be the least upper bound).
)
( n
X 1 n∈N
(a)
k2 k=1
(
(b)
)
n
X
1 n∈N
k k=1
(c)
1
x sin
x
x ∈ (0, 2π)
(d) {x cos (x) | x ∈ [−π, π]}
4. Show that the set of positive real numbers has no smallest element.
5. Find the least upper bound for the following sets
(a) {(−1, 3) ∪ [2, π)}
(b)
y | y = x2 + 2x + 1 | x ∈ (−2, 2)
(c)
(d)
(−1)n
1 n
∈
N
n n
o
2
e−x | x ∈ R
6. Suppose S, T ⊂ R are bounded subsets so that Ls ≤ x ≤ US for all x ∈ S and LT ≤ y ≤ UT for all
y ∈ T . What can you say about bounds for S ∪ T and S ∩ T .
7. Prove that if n is an even integer, then n2 + 2n + 1 is an even integer.