logo for solving-math-problems.com
leftimage for solving-math-problems.com

Mathematics - Properties of Equivalence

by Sagar
(Raipur)










































Let N be a set of natural number, The Symbols, <,<=,= are relations over N. prove or disprove
1.< is reflexive, symmatric or transitive
2.<= is reflexive, symmatric or transitive
3.= is reflexive, symmatric or transitive

Comments for Mathematics - Properties of Equivalence

Click here to add your own comments

Nov 09, 2011
Properties of Equivalence
by: Staff

-------------------------------------------------------------------

Part II

d. Your problem:


Let N be the set of natural numbers. The Symbols, <,<=,= are relations over N. prove or disprove

1.< is reflexive, symmetric or transitive

a = any natural number, b = any natural number, c = any natural number

Reflexive Property test:
Is a < a for all natural numbers?

False - This statement is false for all natural numbers.

For example: 3 < 3 - this statement is not true.

Symmetric Property test:

Does a < b (imply) b < a is true for all natural numbers?

False - This statement is false for all natural numbers.

For example:

3 < 4 (implies) 4 < 3 is not true.

Transitive Property test:

Does a < b and b < c (imply) a < c ?

True - These statements are true for all natural numbers.

For example:

3 < 4 and 4 < 5 (implies) 3 < 5.


2. ≤ is reflexive, symmetric or transitive

a = any natural number, b = any natural number, c = any natural number

Reflexive Property test:

Is a ≤ a for all natural numbers?

True - This statement is true for all natural numbers.

For example: 3 ≤ 3 - this statement is not true.

Symmetric Property test:

Does a ≤ b (imply) b ≤ a is true for all natural numbers?

False - This statement is false for all natural numbers.

For example:

3 ≤ 4 (implies) 4 ≤ 3 is not true.

Transitive Property test:

Does a ≤ b and b ≤ c (imply) a ≤ c ?

True - These statements are true for all natural numbers.

For example:

3 ≤ 4 and 4 ≤ 5 (implies) 3 ≤ 5.



3.= is reflexive, symmetric or transitive


a = any natural number, b = any natural number, c = any natural number

Reflexive Property test:

Does a = a for all natural numbers?

True - This statement is true for all natural numbers.

For example: 3 = 3

Symmetric Property test:

Does a = b and b = a hold true for all natural numbers?

True - These two statements are true for all natural numbers.

For example: 3 = 4 - 1 and 4 - 1 = 3 are both true.

Transitive Property test:

Does a = b and b = c (imply) a = c ?

True - These statements are true for all natural numbers.

For example:

3 = 4 - 1 and 4 - 1 = 5 - 2 (implies) 3 = 5 - 2.

True: all three property tests are true.

The "=" (equal sign) is an equivalence relation for all natural numbers.

This means that the values on either side of the "=" (equal sign) can be substituted for one another.





Thanks for writing.

Staff
www.solving-math-problems.com



Nov 09, 2011
Properties of Equivalence
by: Staff

Part I

Question:

by Sagar
(Raipur)


Let N be a set of natural number, The Symbols, <,<=,= are relations over N. prove or disprove
1.< is reflexive, symmetric or transitive
2.<= is reflexive, symmetric or transitive
3.= is reflexive, symmetric or transitive


Answer:


a. A little background on the Reflexive, Symmetric, and Transitive Properties:

The reason the three properties of equivalence are important is because these properties determine when one algebraic quantity can be substituted for another without changing the original value.

Without being able to substitute equivalent values for one another, many equations could not be solved algebraically.

If an equivalence relation exists, substitution is possible. If two items are equivalent, one item can be replaced with the other, or vice versa.

For example:

The number 3/4 can be replaced with 0.75 to simplify the following arithmetic problem:

9.0367 + 3/4 = ?

Substituting .75 for 3/4

9.0367 + .75 =9.7867


b. All three properties are true for every equivalence relationship:

∙ Reflexive Property

∙ Symmetric Property

∙ Transitive Property

All three properties MUST be considered together.


c. Definitions:


Reflexive Property: (a=a)

a is equivalent to a ("a" equals "a")
The property of reflexivity may seem ridiculously obvious. However, it is used extensively in proofs.


Symmetric Property: (if a=b, then b=a)

The Symmetric and Commutative Property are similar.

a = b is equivalent to b = a

For example:
5+4 = 4+5



Transitive Property: (if a=b and b=c, then a=c)

The Transitive Property illustrates how logic and deductive reasoning are used in mathematics.

The Transitive Property shows how to draw conclusions from the information available.
For example:

.5 = 1/2

1/2 = 2/4

Therefore, .5 = 2/4

-------------------------------------------------------------------

Click here to add your own comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Math Questions & Comments - 01.



Copyright © 2008-2015. All rights reserved. Solving-Math-Problems.com