HW 10 Prof. Caldwell
Due: 17 February 2009 COSC 2300
Decide whether the following are true or false. Say why or why not, in doing so, rely on
the definitions of โand =. Remember that โ
is the same as {}, we use it because in some
contexts it is more readable.
Recall the formal definitions:
AโBdef
=โx.(xโAโxโB)
A=Bdef
=โx.(xโAโxโB)
The informal interpretation of sets as kinds of sacks1in which you can put things may
help with these problems. Under the sack interpretation, the empty set is a sack containing
nothing. The set containing the empty set {{}} (or {โ
}) is a sack that contains an empty
sack. Membership (xโA): means you can reach into the sack Aand pull out an xโ
it does not mean that you can reach in an pull out a sack containing an x, but that you
actually have to be able to grab an xfrom the sack. Subset (AโB), not to be confused
with membership, means that everything that is in the sack Ais also in the sack B,e.g. if
it is possible to pull an xout of Athen you can pull an xout of sack B.Equality: sacks A
and Bare equal if everything that is in Ais also in Band everything that is in Bis also in
A.
You might like to read what Hein has to say on pages 13-15 of chapter one which is linked
on the web-page. Ignore his claim on page 14 that {H, E, L, L, O }is not a set โ otherwise
you might find it useful.
1. 1 โ {1}
2. 1 โ {1}
3. {1} โ {1}
4. โ
โ {1}
5. โ
โ {1}
6. โ
โ โ
7. โ
โ โ
8. {1,2} โ {โ
,1,2}
9. โ
โ {โ
,1,2}
10. โ
โ {1,2}
11. โ
โ {โ
}
12. {โ
} =โ
13. {{โ
}} =โ
1The sack interpretation requires the sacks have the magical property that, if you put the same thing
into the sack more than once and then you take one of them out, theyโre all gone.
1
