Math 360, Fall 2021, 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. Containment (of sets).
2. Equality (of sets).
3. Property.
4. Ordered pair.
5. Cartesian product (of two sets).

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

1. Russell's paradox (this is not really a theorem, but it is an important fact).
2. Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$).

Solve the following problems:

1. Section 0, problems 1, 2, 3, 4, 5, 6, 7, 8, and 11.
2. Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. (Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)
3. Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. (Hint: use the previous result together with the definition of set equality.)
4. The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on Russell's Paradox and its resolutions, see Wikipedia: Universal Set.)

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Solutions:

1. Containment of sets is defined as being contained in a set. There are two ways of being contained. Being contained as elements, using notation $\in \textrm{. Or being contained as a subset, using notations} \subset \textrm{or} \subseteq$. Example: $5 \in \{1, 2, 5\}$ ,$ \{1,2\} \subseteq \{1,2,5\}$, $ 3 \notin \{1,2,5\}$.
2. Equality: two sets are equal if they contain the same elements and have same cardinality. If there are two sets $A \textrm{ and } B$. $A \subseteq B \wedge B \subseteq A \Leftrightarrow A = B$
3. Property is a statement given for describing elements of a set. For example $\{ x| x \textrm{ is even } \wedge x \in \mathbb Z^{+} \wedge x \leqslant 8\}$. However, "x is almost an integer" should not be a property, as it does not well-define a set.
4. Ordered Pair is a pair of objects, the order in this pair is significant. In an ordered pair $(a,b), a \textrm{is the first element and} b \textrm{is the second}. (a,b) \neq (b,a) \textrm{ if } a \neq b.$ For example, $(1,2) = (1,2) \textrm{ because } 1 = 1, 2=2$ and $(1,2) \neq (2,1) \textrm{ because } 1 \neq 2, 2 \neq 1.$
5. Cartesian Product of Two Sets: $\textrm{if } A, B \textrm{ are two sets:}$ $A \times B = \{(a,b) | a \in A \wedge b \in B\}$. For example, $\{1,2\} \times \{3,4\} = \{(1,3),(1,4),(2,3),(2,4)\}$. As a non-example, $\{1,2\} \times \{3,4\} \neq \{(1,2),(3,4)\}$
6. Russell's paradox: if $R \textrm{ is a set and } R = \{x|x \neq x \}$, can one say $R \in R$ or not? Assume that $R \in R, \textrm{ then } R \notin R \textrm{ according to set R's property. However, if } R \notin R, \textrm{ according to R's property, } R \in R$ which creates paradox.
7. Basic Counting Principle: if there are a elements in set A and b elements in set B, there should be $a \times b$ pairs (or elements) in the set which represents Cartesian Product of set A and B. $|A \ times B| = |A| \times |B|$, $(|A|$ is the cardinality of set A).