- Write the negation of each of the following statements (in English,
not symbolically).
- If it rains, then either I will wear a coat or I'll stay home.
- This function has no inverse, and it is not continuous.
- In any triangle, the sum of the measure of the angles is less than .
- For every , there is a so that whenever .
- Every natural number has a unique additive inverse.

- Prove or disprove each of the following statements, using only the
axioms in Appendix 1. Define the set of integers by

As usual, we say is negative if , and is positive if .- For every integer and every integer , is positive and
is negative.
- There are integers and so that is positive and is
negative.
- For every integer , there is an integer so that is
positive and is negative.
- There is an integer so that, for every integer , is positive and is negative.

- For every integer and every integer , is positive and
is negative.
- Consider the following symbolic description of ``kinship''. Our
domain is a set of people, and we have the predicates
- m(x) means ``x is male''.
- f(x) means ``x is female''.
- P(x,y) means ``x is the parent of y''.

- (K1)
- (K2)

- State carefully, in common English, the meaning of axiom K1.
- State carefully, in common English, the meaning of axiom K2.
- Define the predicate to mean . What is the common English meaning of ?
- What is the meaning, in common English, of the assertion ?
- Prove that .

- Prove that that for any natural number , is divisible by
. (Hint: use induction on .)

Scott Sutherland 2002-10-11