Proving/Disproving "Every Pot Has a Lid

[Yankel]

Hello all,

I have a phrase in English, and I need to determine if it is true or false. If it is true, I need to prove it, and if it is false, I need to disprove it.

The phrase is based on the famous phrase "every pot has a lid", and it goes like this:

"If there exist an infinite set of lids, then, all pots has a lid (every pot has a lid)".

As you can see, I have the "if-then" connector here ($$\implies$$), along with the two quantifiers (all and exist). I am not sure how to prove or disprove it.

On one hand, it sounds invalid, since having an infinite number of lids doesn't mean that there is no pot without a lid. On the other hand, the number of pots and number of lids are both natural numbers, and so the cardinality of both is equal, I think.

I was trying to write this down using predicates (using $$\forall \exists \implies \therefore$$), but couldn't do it.

Can you please assist in solving this problem ?

Thank you in advance !

 

[Evgeny.Makarov]

"If there exist an infinite set of lids, then, all pots has a lid (every pot has a lid)".

If I were you, my first reaction would be to refuse to solve this problem. Making mathematical questions in such a way that they essentially involve knowledge about our world of physics, biology and human relationships is a poor way of constructing problems. Yes, problems for younger students often involve names and real situations, e.g., "A teacher announced that she is forming a student theater. Only senior students are eligible, and there has to be at least 14 members...". But such descriptions are readily translated into a purely mathematical model about sets, numbers, functions and so on. If one needs to know what the relationship between a pot and a lid is and how many pots are there in the world (countable and uncountable numbers are both impossible in our physical world), this is not good.

So I would ask you teacher to state the problem using regular mathematical vocabulary. If you have any guesses about the meaning of this phrase, feel free to say.

 

[Yankel]

Thank you, I like your answer. I thought it was only me who thought this problem wasn't phrased properly.

 

[Joppy]

If I were you, my first reaction would be to refuse to solve this problem. Making mathematical questions in such a way that they essentially involve knowledge about our world of physics, biology and human relationships is a poor way of constructing problems. Yes, problems for younger students often involve names and real situations, e.g., "A teacher announced that she is forming a student theater. Only senior students are eligible, and there has to be at least 14 members...". But such descriptions are readily translated into a purely mathematical model about sets, numbers, functions and so on. If one needs to know what the relationship between a pot and a lid is and how many pots are there in the world (countable and uncountable numbers are both impossible in our physical world), this is not good.

So I would ask you teacher to state the problem using regular mathematical vocabulary. If you have any guesses about the meaning of this phrase, feel free to say.

Unfortunately some teachers/lecturers seem to love this sort of ambiguity. I would not be surprised if OP mentioned this to his prof. and got a response of "Oh yeah yeah of course, just assume bla bla and so and so"...