Up A Level "And" "And" of An "Or" Contrapositive "For All" "If and Only If" "If..., Then..." "Not" "Not" of An "And" "Not" of An "If...Then" "Not" of An "Or" "Or" "Or" of An "And" Short Tautologies "There exists"
 
"THERE EXISTS" PHRASES
Variables in mathematics, such as x, y, a, b, c, etc., are quantified with phrases such as
"for all x", "for any z", "for every z", "there is at least one a", "there is some b",
"there is an x", "there is a w", "for at least one j", etc. "There is at least one",
"there is some", "there is an", "there is a", "for at least one", and similar phrases mean
the same thing: there is at least one object with the property being described and MAYBE
more. No claim is being made about there being more than one such object: there might be
exactly one, there might be more than one.

There is a real number x such that, for all real numbers y, x+y=y. This sentence mixes
two kinds of quantifying ("there is" and "for all"). When the "there is" comes first,
it means that there is at least one such object that has some property for ALL of the second
variable under discussion. In this example, x=0 has the property that, for all real numbers
y, 0+y=y.

There is some real number x such that, for all real numbers y, x*y=y. Note that x=1 has
the desired property that, for all real numbers y, 1*y=y.

For all real numbers x, there is a real number y such that x+y=0. This sentence also mixes
two kinds of quantification ("for all" and "there is"). With the "for all" coming first,
the y that is required to exist IS ALLOWED TO BE DIFFERENT FOR EACH x. So, this statement
is true because y=x has the property that x+y=0.

For all real numbers x, there is a real number y such that x*y=1. This sentence is false,
because it happens to have just one exception: when x=0, x*y=0 for all real numbers y and
there is no way to get some y so that 0*y=1.

For all nonzero real numbers x, there is a real number y such that x*y=1. This sentence
is true, because for nonzero x we can let y=1/x. Note that x*(1/x)=1.
 For all positive real numbers x, there is some real number y such that y*y=x. In this example,
there are in fact two such y's: the square root of x and the negative of the square root of x.

For all real numbers x, there is some real number y such that y*y*y=x. In this example,
there is exactly one such ynamely, the real number cube root of x.
 For all E>0, there is some D>0, such that if xc < D then f(x)f(c) < E. This
sentence is equivalent to the continuity of f at c.
Multiple "There Is" Phrases
Consider these two sentences: "There is an x and there is a y such that x+y=y" and "There is a y and
there is an x such that x+y=y". These mean the same thing. Order is not important with multiple
"there is" phrases. This is NOT true for mixtures of "there is" and "for all" phrases.
"Not" Applied To A "There Is" Sentence
Consider what it means for "there is at least one x such that P(x)" to be true, where P(x) is some sentence
about x. For some object x, P(x) is true; maybe only one; maybe more than one. If the sentence is false,
it means that, for every possible x, P(x) is false. So, not[there is at least one x such that P(x)]
means the same as { for all x, [not(P(x))] }.
"For All" Mixed With "There Exists": Order Is Important!
Compare these two sentences: "For all x, there exists at least one y such that Q(x,y)" and "there is at least one y
such that, for all x, Q(x,y)". Here Q(x,y) is some sentence about x and y, such as "x+y=0". The first of these
means that for all x, there is a y WHICH IS ALLOWED TO VARY WITH EACH x which makes Q(x,y) true. So, for example,
"for all x, there exists at least one y such that x+y=0" is true because y=x makes it true.
The sentence "there is at least one y such that, for all x, Q(x,y)" means that there exists one y which does the
job for all x of making Q(x,y) true. This is a much tougher statement to make true! Notice that "there is at least
one y such that, for all x, x+y=0" is false for the real numbersthere y has to be x and must vary with x which
this sentence does not allow.
Of course the tougher statement implies the easier. Consider "there is a z such that, for all x, x+z=x". This is true for real
numbers, because z=0 plays that role for ALL x. It is likewise true that, "for all x, there is at least one z such that x+z=x."
While z is allowed to vary with x to make this last sentence true, it does not have to varyz=0 still works for each and every x!
To summarize this paragraph, the following is always true:
If [there is at least one y such that, for all x, Q(x,y)], then [for all x, there is at least one y such that Q(x,y)].
Warning! The converse is usually false.
