I am currently behind on SLOG posts by a week; It is best to rid of this gap before it gets too large. Here are my reflections on the first and second week of the course.
Thus far, I have personally found the course to be tough. I was never taught highly abstract concepts relating to mathematical logic, like quantifiers ($\forall, \exists$), I only knew how to crunch through numbers and equations.
Sadly, "crunching" through logical claims like:
\[\forall x \in \mathbb {R},\forall y \in \mathbb{R}, x < y \Rightarrow (\exists z \in \mathbb{R}, x < z < y)\]
isn't helpful: Logic needs thought rather than mindless computation.
To increase my comfort with quantifiers, I must first analyse their properties:
Let's take an example and analyze:
These statements look similar - the orders of the quantifiers is just switched! These seemed similar to me at first too, but to my shock I discovered that they have very different meanings...
Let's convert them to english:
$(1)$ means that 'for any integer x there is an integer y that will be greater than x'. We can even further simplify this to 'for every integer there is a integer greater than itself'.
This is a True statement (it would be true...unless we broke math)
$(2)$ says ''there is an integer which is greater than all other integers". This is always False.
See the big difference I was talking about? :D
But how did we arrive at the conclusion that (1) is True and (2) is False?
Well, as I learnt from my 165 lectures, to prove a universal ($\forall$) statement True, we need to show that all elements that obey the statement are True. So, if we set $y$ in $(1)$ to $y = x + 1$, (1) will always be True. Conversely, to prove ($\forall$) statement False, we need to show just one counter example.
Existential ($\exists$) statements are different, to prove one True, we just need to show just one example that obeys the statement. It's like saying "There are stars in the sky" - you just need one star in the sky to make this true. To prove an $\exists$ False, we need to see that all elements in set don't follow the rule. This would happen if there were 'no stars in the sky'.
Here is a table to compare the quantifiers:
\[\forall x \in \mathbb {R},\forall y \in \mathbb{R}, x < y \Rightarrow (\exists z \in \mathbb{R}, x < z < y)\]
isn't helpful: Logic needs thought rather than mindless computation.
To increase my comfort with quantifiers, I must first analyse their properties:
Let's take an example and analyze:
- \[\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z} x < y\]
- \[\exists x \in \mathbb{Z}, \forall y \in \mathbb{Z} x < y\]
These statements look similar - the orders of the quantifiers is just switched! These seemed similar to me at first too, but to my shock I discovered that they have very different meanings...
Let's convert them to english:
$(1)$ means that 'for any integer x there is an integer y that will be greater than x'. We can even further simplify this to 'for every integer there is a integer greater than itself'.
This is a True statement (it would be true...unless we broke math)
$(2)$ says ''there is an integer which is greater than all other integers". This is always False.
See the big difference I was talking about? :D
But how did we arrive at the conclusion that (1) is True and (2) is False?
Well, as I learnt from my 165 lectures, to prove a universal ($\forall$) statement True, we need to show that all elements that obey the statement are True. So, if we set $y$ in $(1)$ to $y = x + 1$, (1) will always be True. Conversely, to prove ($\forall$) statement False, we need to show just one counter example.
Existential ($\exists$) statements are different, to prove one True, we just need to show just one example that obeys the statement. It's like saying "There are stars in the sky" - you just need one star in the sky to make this true. To prove an $\exists$ False, we need to see that all elements in set don't follow the rule. This would happen if there were 'no stars in the sky'.
Here is a table to compare the quantifiers:
| Quantifier | True | False |
| $\forall$ | Show all elements are examples | Show one counter example |
| $\exists$ | Show one example in set | Show all elements are counterexamples |
And I'm done for the day!
Sooham Rafiz
Sooham Rafiz
