Math 360, Fall 2013, Assignment 1

The beginner ... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites.

- P. Halmos

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

1. Cartesian product (of two sets).

   Definition:Cartesian Product

   Fix two sets, A and B. The Cartesian Product of A and B, denoted \(A \ x \ B\), is the set defined as follows:

   \(A \ x \ B = \{ (a,b) \ | \ a\in A \wedge b\in B\}\)

   Example

   Let \(A=\{1,2\}\) and \(B=\{ 3,4 \}\). \(A \ x \ B = \{ (1,3),(1,4),(2,3),(2,4) \}\).

   Non-example

   Let A and B be defined as above. \(A \ x \ B \neq \{ (1,3),(1,4),(1,5),(2,3),(2,4)\}\) because \(5\notin B\).
2. Relation (on a set \(A\)).

   Definition:Relation (on a set \(A\))

   Fix a set A. A relation on A, denoted R, is a subset of the cartesian product \(A \ x \ A\).

   \(R \subseteq A \ x \ A \equiv R \subseteq \{(a,b) \ | a \in A \wedge b \in A \}\)

   Example

   Let \(A=\{1,2\}\), and let R be a set defined as \(R=\{(1,1),(2,1)\}\). \(A \ x \ A = \{(1,1),(1,2),(2,1),(2,2)\}\), thus by inspection, \(R \subseteq A \ x \ A\), and therefore, R is a relation on A.

   Non-example

   Let A be defined as above, but now, let \(R = \{(1,1),(2,3)\}\). By inspection, we observe \(3 \notin A\), so by the definition of the cartesian product, it is impossible that \((2,3) \in A \ x \ A\), thus \(R \not\subseteq A\) and therefore, R is not a relation on A.
3. Reflexive relation.

   Definition:Reflexive

   Fix a relation R on a set A (as defined above). R is said to be reflexive if it is true that:

   \(\forall x\in A (xRx) \equiv \forall x\in A((x,x)\in R)\)

   One way we might say this is: "For all x in the set A, x is related to itself."

   Example

   Let R be the improper subset relation (\(\subseteq\)) on some arbitrary set of sets. Every set is either a subset of itself or equal to itself, therefore the relation is reflexive.

   Non-example

   Now, let R be the strictly less than relation on the set of integers (<). It is not true that every integer is strictly less than itself, therefore the relation is not reflexive.
4. Symmetric relation.

   Definition:Symmetic

   Fix a relation R on a set A (as defined above). R is said to be symmetric if it is true that:

   \(\forall x,y\in A (xRy \to yRx) \equiv \forall x,y\in A((x,y)\in R \to (y,x) \in R)\)

   We say, "For all objects x and y in A, if x is related to y, y is also related to x."

   Example

   Let A = \(\{ x | x \) is a student enrolled in MATH 360\(\}\), and let R be the relationship defined by the statement "x and y are related if(and only if) x and y have the same birthday." Clearly, for any two students x and y, if x has the same birthday as y, y must also have the same birthday as x (you could select ANY two students at random and show this), and thus by the definition of symmetric, R is symmetric.

   Non-example

   Let A = {Square, Rectangle} and let R be the relationship defined by the statement "two elements x and y be related if (and only if), x is a shape that satisfies the definitions of the shape y." A square satisfies all of the requirements of being a rectangle, but a rectangle does not have 4 congruent sides, thus it does not satisfy the requirements of being a square, thus \((square, rectangle) \in R\) but \((rectangle,square)\notin R\) and thus by counterexample, R fails to be symmetric.
5. Transitive relation.

   Definition:Transitive

   Fix relation R on a set A (es defined above). R is said to be transitive if it is true to that:

   \(\forall x,y,z\in A (xRy \and yRz \to xRz) \equiv \forall x,y,z\in A((x,y)\in R \and (y,z)\in R \to (x,z) \in R)\)

   'For all objects x, y, and z in A, if x is related to y and y is also related to z, than x is related to z.'

   Example

   Let \(A = \{2,4,6\} \) and let R be the less than relation. For \((2,4)\) and \((4,6)\) are in the less than relationship R than \((2,6)\) is also in R.

   Non-Example

   Let \(A= \{\{6\}, \{9,8\},\{3,6,9\}\}\). \(6\in \{3,6,9\} and \{3,6,9\} \in \{\{9,8\},\{3,6,9\}\},\) but \(3\notin \{\{9,8\},\{3,6,9\}\}\).

6. Equivalence relation.
7. Partition (of a set).
8. Cell (of a partition).

Solve the following problems:

1. Section 0, problems 1, 5, 7, and 11.

--------------------End of assignment--------------------

Questions:

1.) As a confirmation, because I don't have a hard copy of the book yet, the questions are:

1.) Describe the following set by listing it's elements\[\{x\in\mathbb{R} | x^2=3\}\]

For 5 and 7, decide whether the object described is a set (is well defined). Give an alternative description of each set.

5.) \(\{n\in\mathbb{Z}^+ | n \ is \ a \ large\ number\}\)

7.) \(\{n\in\mathbb{Z} | 39 < n^3 < 57\} \)

11.) List the elements in \(\{a,b,c\} x \{1,2,c\}\)

Thank you for your help in advance. --Robert.Moray (talk) 19:06, 6 September 2013 (EDT)
Yes, those are correct. --Vincent.Luczkow