大學數學的問題 大學數學的問題

Question

(1) write down the contra-positive and converse of the statement if x﹤0
then x^2-x﹥0 and determine (if any) which of the three statements are
true (2) do the same for the statements (a) if x﹥0 then x-x ﹥0
(b)if f(a)﹥0 and f(b)﹤0 then there exists c between a and b such that
f(c)=0 (c)if A and B are convex sets then A∩B is convex
(d) if f(x) is differentiable then f(x) is continuous

什麼叫做convex set

那 (c.1) and (c.2) are True, but (c.3) is not. (EASY) 是怎麼看出來的

Answer 1

註: If P then Q 的 contra-positive 是 If not Q then not P
(contra-positive 和原式的真假值是一樣的)
If P then Q 的 converse 是 If Q then P


(1.1) 原式: if x ﹤0 then x^2-x ﹥0
(1.2) contra-positive: if x^2-x <= 0 then x >= 0
(1.3) converse: if x^2-x ﹥0 then x ﹤0
(1.1) and (1.2) are TRUE (EASY).
(1.3) is not true, because x > 1 also makes x^2-x ﹥0

(a.1) 原式: if x ﹥0 then x-x ﹥0
(a.2) contra-positive: if x-x <= 0 then x <= 0
(a.3) converse: if x-x ﹥0 then x > 0
(a.1) and (a.2) are False and (a.3) is True, because x-x is zero.

待續...

2007-09-18 00:57:18 補充：
(b.1) 原式: if f(a)﹥0 and f(b)﹤0 then
there exists c between a and b such that f(c)=0
(b.2) contra-positive: if there is no c between a and b such that f(c)=0 then
f(a) <= 0 or f(b) >= 0

2007-09-18 01:01:51 補充：
(b.3) converse: if there exists c between a and b such that f(c)=0 then
f(a)﹥0 and f(b)﹤0
(b.1) and (b.2) are False, because f(x) has to be continuous--
this is called the Intermediate Theorem (中間值定理) in CALCULUS.

2007-09-18 01:02:37 補充：
(b.3) is false, because there are other cases.

(c.1) 原式: if A and B are convex sets then A∩B is convex
(c.2) contra-positive: if A∩B is not a convex set then
A or B is not convex

2007-09-18 01:03:18 補充：
(c.3) converse: If A∩B is a convex set, then A and B are convex.
(c.1) and (c.2) are True, but (c.3) is not. (EASY)

2007-09-18 01:04:31 補充：
(d.1) if f is differentiable then f is continuous
(d.2) if f is discontinuous then f is not differentiable.
(d.3) If f is continuous then f is differentiable

2007-09-18 01:05:34 補充：
(d.1) and (d.2) are TRUE--an important theorem (可微則連續) in CALCULUS!!
(d.3) is FALSE-- |x| is conti. at 0, but not diff. at 0.

2007-09-18 09:10:37 補充：
convex set?
說來話長,
簡單的講, 您就把 convex set 當做是凸多邊形就好了.