Is this symbolic expression correct?

Question

Say C: set of courses

P(x,y): 'x is a prerequisite for course y'

statement: 'some courses have the same prerequisites'

Is this symbolic expression correct?

If not, how would I write this with implication? Also how would I write this without implication?

∃ x ∈ C, ∃ y ∈ C, ∃ z ∈ C, P(y, x) ∧ P(z, x) ∧ y = z

Thanks again!

Answer 1

You wrote something like this:

Some course has at least one requisite and that requisite at least sometimes equals itself even if I try to use a different variable to denote it.

Try to translate the following to formal language:

There exists two distinct courses such that any course is prerequisite of the first if and only if it is prerequisite of the other.

Answer 2

What you’ve written simply says that there is a course with a prerequisite (that you happen to have mentioned twice, once as $y$ and once as $z$). First we need to say that there are (at least) two courses:

$$\exists x\in C\,\exists y\in C(x\ne y)\;.$$

Now we need to add something to that to say that these courses $x$ and $y$ have a prerequisite in common; I’ve included an answer, but I’ve left it spoiler-protected.

$$\exists x\in C\,\exists y\in C\,\exists z\in C\Big(x\ne y\land C(z,x)\land C(z,y)\Big)$$

My answer does not use an implication, and I don’t think that there is a really natural way of saying this that does use an implication.