Assignment

Due on Monday, September 29th.

Part 1

Taken from Partee, Meulen, and Wall (1990).

Consider the following sets:

\[S_{1} = \{\{\varnothing\}, \{A\}, A\}\] \[S_{2} = A\] \[S_{3} = \{A\}\] \[S_{4} = \{\{A\}\}\] \[S_{5} = \{\{A\}, A\}\] \[S_{6} = \varnothing\] \[S_{7} = \{\varnothing\}\] \[S_{8} = \{\{\varnothing\}\}\] \[S_{9} = \{\varnothing, \{\varnothing\}\}\]

• Of the sets \(S_{1}\) through \(S_{9}\), which are members of \(S_{1}\)?
• Which are subsets of \(S_{1}\)?
• Which are members of \(S_{9}\)?
• Which are subsets of \(S_{9}\)?
• Which are members of \(S_{4}\)?
• Which are subsets of \(S_{4}\)?

Part 2

Show that \(A ∩ B ⊆ A\) for any arbitrary sets \(A\) and \(B\).

Part 3

Show that if \(A ⊆ B\), then \(A = A ∩ B\) for any arbitrary sets \(A\) and \(B\).

References

Partee, Barbara H., Alice ter Meulen, and Robert E. Wall. 1990. Mathematical Methods in Linguistics. Vol. 30. Studies in Linguistics and Philosophy. Dordrecht: Kluwer Academic Publishers.