Math 360, Fall 2020, 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.
2. Section 6 only of A short introduction to $\LaTeX$
3. Skim through the List of $\LaTeX$ mathematical symbols.

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).
6. Binary relation (from $A$ to $B$).
7. Bubble diagram (to visualize a binary relation).

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

1. Russel'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. Click the "edit" link on this page. This will show you the "source code" that produces the mathematical typesetting in this assignment. Then gather your courage, and try asking some mathematical questions in the "Questions" section below.
3. Really, you will have a much easier time on quizzes and tests if you begin to pick up some basic $\LaTeX$ right now. For more examples, click on the "Main page" link at the left, and begin looking through the assignment pages for some of my old courses. Math 360, Fall 2019 would be a reasonable place to start. Please do not actually edit the old course pages, but click on the "edit" link to see the source code, then back out.
4. Examine the source code that produces the following: (i) $\frac{d}{dx}\left[x^2+3x+4\right] = 2x+3$, (ii) $\int_0^1 e^x\,dx = \left.e^x\right|_0^1=e-1 \approx 1.71$, and (iii) if $(S,\triangle)$ is a binary structure with two-sided identity element $e$, and $s\in S$ is arbitrary, then $e\triangle s = s\triangle e = s$. Google around until you understand what's going on.
5. Make a new section in this assignment page called "Scratch Work" and experiment with basic $\LaTeX$ to your heart's content. This is a good investment of your time, and your future self will thank you for it.

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Solutions:

1. Containment (of sets).

S and T are sets.

If one set is contained, ⊆ in another means that every element, ∈ of S is also a member of T.

"S ∈ T" means whenever "S is contained in T"

x ∈ S, also x ∈ T means elements or members of S are elements of T.

S = {4, 5, 6}

T = {4, 5, 6, 7}

S ⊆ T , T ≠ S

2. Equality (of sets).

If S is equal to T, S=T means S ⊆ T and T ⊆ S, mutual containment. Two sets are the same.

S = { 6, 7, 8}

T = {6, 6, 7, 8}

S ⊆ T, T ⊆ S therefore, T=S

Listing a element twice gives no meaning when set building.

3. Property.

A property is a statement that is either true or false of any given object.

P(x) =

4. Ordered pair.

A pair of two integers

5. Cartesian product (of two sets).

6. Binary relation (from A to B).

7. Bubble diagram (to visualize a binary relation).

Solution 0

1. The set of all x's that are a member of the real numbers, such that x squared is three.

√($x^2$) = -√3 , +√3

x = {-√3, +√3}

2. The set of all m's that are a member of the integers, such that m squared is three.

x = { }

3. The set of all m's that are a member of the integers, such that m*n = 60, for some n is a member of integers.

{ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, -1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30, -60}

4. The set of all m's that are a member of the integers, such that m$^2$-m<115}

{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

5. The largest integer is positive infinity so, I think it's null.

6. Every square is greater than 0. Therefore, null.

7.

3^3$=27, 4^3$=64.

Therefore, null.

8. A irrational number is not a integer, not well-defined.

11. {(a,1), (a,2), (a,c), (b,1), (b,2), (b,c), (c,1), (c,2), (c,c)}