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Material Type: Assignment; Professor: Caldwell; Class: Discrete Structures; Subject: Computer Science; University: University of Wyoming; Term: Spring 2009;
Typology: Assignments
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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 โ and =. Remember that โ is the same as {}, we use it because in some contexts it is more readable. Recall the formal definitions:
A โ B def = โx.(x โ A โ x โ B) A = B def = โx.(x โ A โ x โ B)
The informal interpretation of sets as kinds of sacks^1 in 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 ( x โ A): means you can reach into the sack A and pull out an x โ 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 x from the sack. Subset (A โ B), not to be confused with membership, means that everything that is in the sack A is also in the sack B, e.g. if it is possible to pull an x out of A then you can pull an x out of sack B. Equality: sacks A and B are equal if everything that is in A is also in B and everything that is in B is 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.
into the sack more than once and then you take one of them out, theyโre all gone.