Elimination of quantifiers

Ok kiddo, let's say you have a bunch of things like apples, oranges and bananas, and you want to say something about them, like "there are more apples than bananas". But instead of using words like "more" or "less", we can use math symbols like the greater than and less than signs to make it easier.

Now let's say you have a bunch of these math statements and you want to find out if they are true or false. We can use a fancy thing called quantifiers to do this. Quantifiers are like special words that tell you if something is true for all the things you are talking about, or if it's only true for some of them.

For example, let's say you have a statement like "there is at least one apple". We can use a fancy symbol called the existential quantifier (∃) to show that this statement is true for at least one thing in our group of apples, oranges and bananas.

But what if we have a statement like "there are more apples than bananas"? We need to use another type of quantifier called the universal quantifier (∀) to show that this statement is true for all the things in our group.

Now let's say we want to get rid of these quantifiers and change our statement into a simpler math statement. We can do this by using something called "elimination of quantifiers". Basically, we use some math rules to rewrite our statement without the fancy symbols.

For example, if we have a statement like "∃x (x is an apple)", which means "there is at least one apple", we can get rid of the existential quantifier (∃) by using something called an "existential instantiation". This just means we pick one thing from our group (say, an apple) and substitute it for the variable x. So now our statement becomes "an apple is an apple", which is just a fancy way of saying "there is at least one apple".

Similarly, if we have a statement like "∀x (x is an apple → ∃y (y is a banana ∧ y is less than x))", which means "for all apples, there is at least one banana that is less than it", we can get rid of the universal quantifier (∀) by using something called "universal instantiation". This just means we pick any apple (let's call it A) and substitute it for the variable x. Then we use "existential instantiation" to pick a banana (let's call it B) that is less than A. So now our statement becomes "for any apple A, there is a banana B such that B is less than A", which is just a fancy way of saying "there is always a banana that is smaller than an apple in our group".

So that's it kiddo, "elimination of quantifiers" just means we use some math rules to rewrite fancy statements without the special words. It's like taking off their fancy clothing and seeing that they're really just basic math statements underneath.