Exercises for Week 2

Unless otherwise specified, the axioms that you should use for these exercises are the ones that have been introduced so far, namely Extensionality, Separation, Pairing, Union, Infinity, and Replacement. Assume that "relation" means "binary relation" unless otherwise specified.

1. Without using Replacement, prove that if a relation $R$ is a set then its domain $\text{dom}(R)$ and range $\text{ran}(R)$ are also sets. Hint: You must use the actual definition of Kuratowski ordered pairs and not just the property from Exercise 2 of last week.
2. Let (*) be the axiom schema saying that if a class $F$ is a function whose domain $\text{dom}(F)$ is a set, then the function $F$ itself is a set. Prove that (*) is equivalent to replacement (in the context of the other axioms that have been introduced so far.)
3. Prove that if $A$ and $B$ are sets, then the Cartesian product $A \times B$ is a set. Hint: first consider the special case where $A = \{x\}$.
4. Prove or disprove each of the following statements about ordinal arithmetic.
   1. Multiplication is left-distributive over addition.
   2. Multiplication is right-distributive over addition.
5. Prove that ordinal multiplication is associative.