## 80-110 The Nature of Mathematical Reasoning

**Spring 2001**

*Dirk Schlimm*
# Homework No.8

Thursday, March 15, 2001

Due Tuesday, March 20, 2001

**1. Probability theory** (4 points)
a) Prove Bayes' Theorem: P(H|E)=[ P(E|H)*P(H) ] / P(E)

from the axioms and definitions introduced in class.

b) Assume that H stands for "The hypothesis is true" and E stands for
"There is evidence that supports the hypothesis". Under this
interpretation, what does P(E|H) mean? What probabilities do you need
to know, in order to determine the value for the probability that the
hypothesis is true given that there is evidence for it?

**2. Syllogisms** (3 points)

Read Handout #20. Do the following problems:

- Page 49, Problems 1 and 2.
- Page 52, Problem 1.

**3. The Language of First-Order Logic** (3 points)

In FOL read,

- Section 2.8, and
- Sections 3.1-3.4.

Do the problems listed below. Hand in your answers on
a disk (in PC/IBM format). Name your files 3-6.WLD, 3-7-1.WLD, and
3-7-2.WLD
- Page 44, Problem 6.
- Page 44, Problem 7.