I have the following Hasse diagram below, the question is given specific generalised quantifiers I have to list the subsets of {a,b,c,d} which the quantifier corresponds to.
I have completed the question and would like to check if my answers make sense.
Generalised quantifier subsets
1. Every {b,c} bc,abc,bcd,abcd
2. At least two {a,b,c} abc,ab,ac,bc,abd,acd,bcd,abcd
3. Fewer than two {a,b,c,d} a,b,c,d, {} <- empty set
4. At most two {b,d} bd,b,d,cd,bc,ad,ab,abc,acd,abcd, {}
5. Exactly two {a,b,c} ab,bc,ac,abd,acd,bcd
6. Some but not all {d,a,c} d,a,c,acd,ac,ad,ab,bd,bc
7. Ic c
8. Exactly as many {a,b,c} as {b,d} b
9. Id *OR* NO {c,b} d,ad
10. Ic *AND* NOT Id c,ac,bc,abc
I am not too sure about 6,7,8,9 and 10. Regarding question 6 I understand it as a proper subset of {a,b,c,d} when comparing it with {d,a,c}.
Questions 7, 9 and 10, symbol I is as far as I understand an Identity relation (could not find much on this except that its basically just itself? as in Id would return a subset of just {d}). In saying that 9 and 10 then contradict to my understanding of it and are most likely incorrect.
Finally for question 4 I added the empty set {} because the question asks "at most two" so I figured it could be (nothing, some combination or single element of the subset given)
Some background on generalised quantifiers: http://en.wikipedia.org/wiki/Generalized_quantifier
Thank you
If I’m interpreting everything correctly, $1,2,3$, and $5$ are correct.
$4$ is not: the set $\{b,d\}$ has only two elements, so every subset of $\{a,b,c,d\}$ contains at most two elements from $\{b,d\}$. Thus, you’re missing $a,c,ac,abd$, and $bcd$.
$6$ is also wrong: you should not have $acd$, as it has all three of $d,a$, and $c$. The only sets that you may not have are $\{\},b,acd$, and $abcd$, so the correct list is: $a,c,d,ac$, $ad,cd,ab,bc,bd,abc,abd,bcd$.
$7$ is correct if your understanding of $I$ is correct.
In $8$ you’ve missed quite a few possibilities: $\{\}$ has $0$ from each of $\{a,b,c\}$ and $\{b,d\}$; $ad$ and $cd$, as well as $b$, have one from each of those sets; and $abd$ and $bcd$ have $2$ from each of those sets.
You’ve also missed several in $9$: if $\text{NO }\{c,d\}$ means ‘nothing from the set $\{c,d\}$’, the list should also include $a$ and $\{\}$. (Here again I’m assuming that your interpretation of $I$ is correct.)
Finally, the only subset of $\{a,b,c,d\}$ that is both identical to $\{c\}$ and not identical to $\{d\}$ is $\{c\}$, so the only correct item on your list is $c$.