2. Define the following concepts using the symbols ‘∀’, ‘s.t.’, ‘and’, ‘∈’, ‘/∈
’, ‘∃’ (without
using ‘/∃’):
(a) f is not uniformly continuous on D.
(b) f is not differentiable at x0.
’, ‘∃’ (without
using ‘/∃’):
(a) f is not uniformly continuous on D.
(b) f is not differentiable at x0.
-
I assume we can also use basic symbols involving real numbers (e.g. the notion of absolute value and the ordering <).
In (a),
[there exists] e > 0, [for all] d > 0, [there exist] x, y in D, |x - y| < d and |f(x) - f(y)| >= e.
In (b) (assuming f is real valued, and not taking values in some more complicated vector space)
[for all] L, [there exists] e > 0, [for all] d > 0, [there exists] x in D, 0 < |x - x_0| < d and |(f(x) - f(x_0))/(x - x_0) - L| >= e.
(If it helps in interpreting (b): to say that f is not differentiable at x_0 is to say that lim_{x to x_0} (f(x) - f(x_0))/(x - x_0) does not exist, which is to say that for all real numbers L, it is not the case that lim_{x to x_0} (f(x) - f(x_0))/(x - x_0) = L. This last statement is what the above symbols convey.
The "s.t." symbol is not generally used in formal treatments of symbolic logic. The main reason for this is that if you pay attention to the order of quantifiers--- as you have to anyway, in formal treatments of symbolic logic--- it is completely redundant. For example, both of the above sentences are completely unambiguous without "s.t."s. To give an idea of the redundancy, imagine that someone invented a "i.i.t.c.t." symbol for "it is the case that", and they inserted it after every for all statement--- so they wrote for example "[for all] L > 0, i.i.t.c.t. _____" instead of "[for all] L > 0, _____". It should be clear that the "i.i.t.c.t." is not actually needed; if you put it there, its only role is to make the symbolic sentence a little closer to English. Well, it's exactly the same with "s.t.".
You really only find "s.t." used as a "logical symbol" in textbooks that introduce some notions from symbolic logic, without ever getting serious about it. The idea, I think, is that putting "s.t."s in makes symbolic sentences easier to read. I'm not sure I agree with this, but in any case, if you wanted to put some "s.t."'s in, you certainly could:
[there exists] epsilon > 0 s.t. [forall] delta > 0, [there exist] x, y in D s.t. |x - y| < d and |f(x) - f(y)| >= e.
[for all] L, [there exists] e > 0 s.t. [for all] d > 0, [there exists] x in D s.t. 0 < |x - x_0| < d and |(f(x) - f(x_0))/(x - x_0) - L| >= e.
In (a),
[there exists] e > 0, [for all] d > 0, [there exist] x, y in D, |x - y| < d and |f(x) - f(y)| >= e.
In (b) (assuming f is real valued, and not taking values in some more complicated vector space)
[for all] L, [there exists] e > 0, [for all] d > 0, [there exists] x in D, 0 < |x - x_0| < d and |(f(x) - f(x_0))/(x - x_0) - L| >= e.
(If it helps in interpreting (b): to say that f is not differentiable at x_0 is to say that lim_{x to x_0} (f(x) - f(x_0))/(x - x_0) does not exist, which is to say that for all real numbers L, it is not the case that lim_{x to x_0} (f(x) - f(x_0))/(x - x_0) = L. This last statement is what the above symbols convey.
The "s.t." symbol is not generally used in formal treatments of symbolic logic. The main reason for this is that if you pay attention to the order of quantifiers--- as you have to anyway, in formal treatments of symbolic logic--- it is completely redundant. For example, both of the above sentences are completely unambiguous without "s.t."s. To give an idea of the redundancy, imagine that someone invented a "i.i.t.c.t." symbol for "it is the case that", and they inserted it after every for all statement--- so they wrote for example "[for all] L > 0, i.i.t.c.t. _____" instead of "[for all] L > 0, _____". It should be clear that the "i.i.t.c.t." is not actually needed; if you put it there, its only role is to make the symbolic sentence a little closer to English. Well, it's exactly the same with "s.t.".
You really only find "s.t." used as a "logical symbol" in textbooks that introduce some notions from symbolic logic, without ever getting serious about it. The idea, I think, is that putting "s.t."s in makes symbolic sentences easier to read. I'm not sure I agree with this, but in any case, if you wanted to put some "s.t."'s in, you certainly could:
[there exists] epsilon > 0 s.t. [forall] delta > 0, [there exist] x, y in D s.t. |x - y| < d and |f(x) - f(y)| >= e.
[for all] L, [there exists] e > 0 s.t. [for all] d > 0, [there exists] x in D s.t. 0 < |x - x_0| < d and |(f(x) - f(x_0))/(x - x_0) - L| >= e.