Difference between revisions of "Math 360, Fall 2013, Assignment 1"

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==Carefully define the following terms, then give one example and one non-example of each:==
  
==Carefully define the following terms, then give one example and one non-example of each:==
   
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# Cartesian product (of two sets).<p>'''Definition:'''Cartesian Product</p><p>''Fix two sets, ''A'' and ''B''. The '''Cartesian Product ''A'' and ''B''''', denoted <math>A \ x \ B</math>, is the set defined as follows:''</p><p><math>A \ x \ B = \{ (a,b) \ | \ a\in A \wedge b\in B\}</math></p><p>'''Example'''</p> Let <math>A=\{1,2\}</math> and <math>B=\{ 3,4 \}</math>. <math>A \ x \ B = \{ (1,3),(1,4),(2,3),(2,4) \}</math>.<p>'''Non-example'''</p> Let ''A'' and ''B'' be defined as above. <math>A \ x \ B \neq \{ (1,3),(1,4),(1,5),(2,3),(2,4)\}</math> because <math>5\notin B</math>
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# Cartesian product (of two sets).<p>'''Definition:'''Cartesian Product</p><p>''Fix two sets, ''A'' and ''B''. The '''Cartesian Product ''A'' and ''B''''', denoted <math>A \ x \ B</math>, is the set defined as follows:''</p><p><math>A \ x \ B = \{ (a,b) \ | \ a\in A \wedge b\in B\}</math></p><p>'''Example'''</p> Let <math>A=\{1,2\}</math> and <math>B=\{ 3,4 \}</math>. <math>A \ x \ B = \{ (1,3),(1,4),(2,3),(2,4) \}</math>.<p>'''Non-example'''</p> Let ''A'' and ''B'' be defined as above. <math>A \ x \ B \neq \{ (1,3),(1,4),(1,5),(2,3),(2,4)\}</math> because <math>5\notin B</math>.
 
# Relation (on a set <math>A</math>).
  
# Relation (on a set <math>A</math>).
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# Reflexive relation.<p>'''Definition:'''Reflexive</p><p>''Fix a relation R on a set A (as defined above). R is said to be '''reflexive''' if it is true that:</p><p><math>\forall x\in A (xRx) \equiv \forall x\in A((x,x)\in R)</math></p><p>'''Example'''</p> Let R be the improper subset relation (<math>\subseteq</math>) on some arbitrary set of sets. Every set is either a subset of itself or equal to itself, therefore the relation is reflexive<p>'''Non-example'''</p> Now, let R be the strictly less than relationship on the set of integers (<). It is not true that every integer is less than itself, therefore the relation is not reflexive.
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# Reflexive relation.<p>'''Definition:'''Reflexive</p><p>''Fix a relation R on a set A (as defined above). R is said to be '''reflexive''' if it is true that:</p><p><math>\forall x\in A (xRx) \equiv \forall x\in A((x,x)\in R)</math></p><p>'''Example'''</p> Let R be the improper subset relation (<math>\subseteq</math>) on some arbitrary set of sets. Every set is either a subset of itself or equal to itself, therefore the relation is reflexive.<p>'''Non-example'''</p> Now, let R be the strictly less than relationship on the set of integers (<). It is not true that every integer is less than itself, therefore the relation is not reflexive.
 
# Symmetric relation.
  
# Symmetric relation.
 
# Transitive relation.
  
# Transitive relation.

Revision as of 17:00, 7 September 2013

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 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 \(A\)).
  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:

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

    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 relationship on the set of integers (<). It is not true that every integer is less than itself, therefore the relation is not reflexive.
  4. Symmetric relation.
  5. Transitive relation.
  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

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