Sets Notation

Actually, there is no good in just knowing the answer. I hope you treat the following answers as a reference for checking.

A1) (-inf,-1)U(0,1)
-,-,- or +,-,+

A2) (-inf,M)U(M,inf)
For inf, no "(" or ")"

A3) (-2,-1)U[1,2]
Here ")" for -1 as excludes "[-1"
Similarly, "[" for 1 as excludes "1)"

A4) phi (empty set) / {}

B1) Sup(S) = 1/1! = 1 ; inf(S) = 0

B2) S = (2,5)
Sup(S) = 5 ; inf(S) = 2
Even though 2 (/ 5) does not belong to S, it can be the infimum (/ supremum).

B3) S = {0,1}
Sup(S) = 1; Inf(S) = 0

B4)
.Inf(S) = 1 - 1/2^1 = 1/2
.Because for a < b, 1 - 1/2^a < 1 - 1/2^b
.Sup(S) = 1
.It suffices to show that the sequence 1 - 1/(2^n) is increasing and converges to 1.
...Increasing)
......for a < b, 1 - 1/2^a < 1 - 1/2^b
...Bounded by 1)
......-1/(2^n) < 0 => 1 - 1/(2^n) < 1
...Approaching 1)
......For all epsilon > 0, we can find a sufficiently large natural number M > ln(epsilon)/ln(1/2) such that | 1 - (1-1/2^M) | = 1/2^M < epsilon.
.Hence, the proposition stands.

2011-09-07 14:41:40 補充：
001 wrong at A3

Wish it helps


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