Suppose we have the following open statements on x:

$\ S(x) $ means that "x is a student"

$\ P(x) $ means that "x likes pizza"

$\ L(x,y) $ means that "x loves y"

$\ K(x,y) $ means that "x knows y"

How would we write the following statements?

  1. All students like pizza
  2. Everyone has somebody who loves them
  3. Only students like pizza
  4. All love is mutual
  5. Everyone likes pizza or everyone is a student
  6. I know someone who knows someone who knows Ariana Grande
  7. Everyone who knows Ariana Grande loves her

Whats mainly confusing me is the separation of these variables. Some with 1 and the other "connectors" with 2. How would I properly answer these?


I'll just do 2 as an example ... Hopefully that'll help you with the others.

Everyone has somebody who loves them.

OK, so you need a universal for 'everyone', but you also an existential for the 'someone' who loves them. So:

$$\forall x \exists y L(y,x)$$

  • $\begingroup$ How would you write something like "all love is mutual" $\endgroup$ – 1011011010010100011 Jun 22 '17 at 21:01
  • $\begingroup$ Also, how would you write the first one? Vx(S(x)->P(x))? $\endgroup$ – 1011011010010100011 Jun 22 '17 at 21:02
  • $\begingroup$ @Mitchell Yes, that first one is correct. The 'all love is mutual' means that if $x$ loves $y$, then $y$ loves $x$. Why don't you post your attempts for all these, and then you can get some feedback ... $\endgroup$ – Bram28 Jun 22 '17 at 21:23
  • $\begingroup$ Also, use mathjax. $\forall x~(S(x)\to P(x))$ is written $\forall x~(S(x)\to P(x))$ and $\forall x~\exists y~L(y,x)$ is $\forall x~\exists y~L(y,x)$ $\endgroup$ – Graham Kemp Jun 23 '17 at 1:04

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