Math 360, Fall 2017, Assignment 1

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

Read:

1. Section 0.

Carefully define the following terms, then give one example and one non-example of each:

1. Equality, set, membership, and property. (Strictly speaking, these words are not defined, but one can still give examples and non-examples.)
2. Subset.
3. Cartesian product (of two sets).
4. Relation (from one set to another).
5. Reflexive (relation from $A$ to $A$).
6. Symmetric (relation from $A$ to $A$).
7. Transitive (relation from $A$ to $A$).

Carefully state the following theorems (you do not need to prove them):

1. Axiom of equality (this is an axiom, not a theorem, but nevertheless it provides the criterion by which we establish that two sets are equal).
2. Criterion for equality of ordered pairs.
3. Russell's paradox (this is not exactly a theorem, but it is an important fact).

Solve the following problems:

1. Section 0, problems 1, 3, 5, 7, 11, 29, 30, 31, 32, and 36(a). (In problems 29-35, an equivalence relation is a relation which is reflexive, symmetric, and transitive. Please ignore the parts of these questions that refer to partitions; we will talk about these next week.)

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