  ## Math Properties:examplesofReflexive, Symmetric,andTransitiveEquivalence Properties Math Properties - Equivalence Relations

Properties of Real Numbers :

Real numbers have unique properties which make them particularly useful in everyday life.

First , Real numbers are an ordered set of numbers. This means real numbers are sequential. The numerical value of every real number fits between the numerical values two other real numbers.

Everyone is familiar with this idea since all measurements (weight, the purchasing power of money, the speed of a car, etc.) depend upon the fact that some numbers have a higher value than other numbers. Ten is greater than five, and five is greater than four . . . and so on. This is an important math property.

Second , we never run out of real numbers. The quantity of real numbers available is not fixed. There are an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite "row of separate points" on a number line. There is always another real number whose value falls between any two real numbers (this is called the "density" property).

Third , when real numbers are added or multiplied, the result is always another real number (this is called the "closure" property). [This is not the case with all arithmetic operations. For example, the square root of a -1 yields an imaginary number.]

With these three points in mind, the question is: , How can we use real numbers in practical calculations? What rules apply?

• How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?

• Does it matter what we do first? second? third? . . .

• Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

• Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

The following properties of real numbers answer these types of questions. The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

• Associative Property

• Commutative Property

• Distributive Property

• Identity Property

• Inverse Property

In addition, the following three math properties of equivalence determine when one algebraic quantity can be substituted for another without changing the original value.

• Reflexive Property

• Symmetric Property

• Transitive Property

Properties of Equivalence - click description "Math Properties"  ; color:#0000FF; "Top of Page" Examples of Reflexive, Symmetric, and Transitive Equivalence Properties

An Equivalence Relationship always satisfies three conditions:

• Reflexive Property

• Symmetric Property

• Transitive Property Is the "=" (the equal sign) an equivalence relation for all real numbers?

a = any real number, b = any real number, c = any real number

Reflexive Property test:

Does a = a for all real numbers?

True - This statement is true for all real numbers.

For example: 3 = 3

Symmetric Property test:

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

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

This means that the values on either side of the "=" (equal sign) can be substituted for one another. Is the ">" (the greater than symbol) an equivalence relation for all real numbers?

a = any real number, b = any real number, c = any real number

Reflexive Property test:

Is a > a for all real numbers?

False - This statement is false for all real numbers.

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

Symmetric Property test:

Does a > b (imply) b > a is true for all real numbers?

False - This statement is false for all real numbers.

For example:

3 > 2 (implies) 2 > 3 is not true.

Transitive Property test:

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

True - These statements are true for all real numbers.

For example:

3 > 2 and 2 > 1 (implies) 3 > 1.

False: two of the three property tests are false .

The ">" (greater than symbol) is not an equivalence relation for all real numbers.

This means that the values on either side of the ">" (greater than symbol) cannot be substituted for one another. Is the " " (the greater than or equal to symbol) an equivalence relation for all real numbers?

a = any real number, b = any real number, c = any real number

Reflexive Property test:

Is a a for all real numbers?

True - This statement is true for all real numbers.

For example: 3 3 - this statement is true.

Symmetric Property test:

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

False - This statement is false for all real numbers.

For example:

3 2 (implies) 2 3 is not true.

Transitive Property test:

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

True - These statements are true for all real numbers.

For example:

3 2 and 2 1 (implies) 3 1.

False: one of the three property tests is false .

The " " (greater than or equal to symbol) is not an equivalence relation for all real numbers.

This means that the values on either side of the " " (greater than or equal to symbol) cannot be substituted for one another.