# Exercise 7.72#

Let \(π\) be the sheaf of people as in Section 7.4.3, and let \(π: \Omega β \Omega\) be βassuming Bob is in San Diego β¦β

Name any predicate \(π: π β \Omega\), such as βlikes the weather.β

Choose a time interval \(π\). For an arbitrary person \(π β π(π)\), what sort of thing is \(π(π )\), and what does it mean?

What sort of thing is \(π(π(π ))\) and what does it mean?

Is it true that \(π(π ) β€ π(π(π ))\)? Explain briefly.

Is it true that \(π(π(π(π ))) = π(π(π ))\)? Explain briefly.

Choose another predicate \(π: π β \Omega\). Is it true that \(π(π β§ π) = π(π) β§ π(π)\)? Explain briefly.

## Alternative answer#

Letβs use βlikes the weatherβ as suggested.

Weβll let \(s\) equal Bob to make this all simpler, although we could write it all for an arbitrary person. So \(p(s)\) is a particular subset of times Bob likes the weather.

We translate this predicate to βBob likes the weather or he is not in San Diegoβ using the βor notβ translation discussed elsewhere. We expect this sheaf to return a subset of times, and this predicate should be able to return a subset of times.

Or more precisely \(π(π ) β€_S π(π(π ))\). While \(p(s)\) is a particular set of times not qualified by anything, \(j(p(s))\) is qualified by the requirement that Bob not be in San Diego. Most precisely, the subset of times that βBob likes the weatherβ given by \(p(s)\) should be a subset of the times that itβs true that βBob likes the weather or he is not in San Diegoβ.

Yes; adding an additional qualifier of βBob is in San Diegoβ should make no difference because weβve already qualified the statement on that proposition. If Bob was not in San Diego, heβs still not in San Diego.

Letβs assign \(q\) to ββ¦ is wearing redβ. Then βBob both likes the weather and is wearing red, and is not in San Diegoβ should be equivalent to βBob likes the weather and is not in San Diego, and Bob is wearing red and is not in San Diegoβ.