What is the difference between these two statements?

Question

I was asked in a course of mine to construct a set where only one of the following statements holds. The statements are:

a) For every yellow number a there is a blue number b such that $a<b$

b) There is a blue number d such that for every yellow number c, it is true that $c<d$.

To me, they look equivalent. How are they different?

Answer 1

a) For every man $a$ there is a woman $b$ such that $b$ is the mother of $a$.

b) There is a woman $d$ such that for every man $c$, $d$ is the mother of $c$.

The first says that every man has a mother. This is true.

The second says that there is one woman who is every man's mother, that every man has the same mother. This is false.

This is because when we interpret quantifiers (like “for every” and “there is”) we understand that the choices in the later quantifiers can depend on the choices in the earlier quantifiers, but not the other way around.

So the identity of woman $b$ can depend on the choice of the man $a$, and we can have a different $b$ for each $a$.

But the identity of the woman $d$ cannot depend on the choice of the man $c$. It must be the same $d$ for every $c$.

Answer 2

Consider the natural numbers and eplace blue with even and yellow with odd. Then only one of the two statemens is true:

a) Given yellow $a$, you can pick blue $b=a+1$

b) Assume $d$ is such a blue number; then the claim fails for the yellow number $c=d+1$.

Answer 3

Statement $<b>$ is stronger than statement $<a>$:

• Statement $<a>$: For every yellow number, there is a larger blue number.

• Statement $<b>$: There is ONE blue number larger than ALL yellow numbers.

So you can say that $<b>\implies<a>$, but you cannot say that $<a>\implies<b>$.