  ## Math Properties . . . Equivalence: Reflexive, Symmetric, and Transitive 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"   "Top of Page"

What guarantees that a relationship is an Equivalence Relationship ?

The following three properties are true for every equivalence relationship:

• Reflexive Property

• Symmetric Property

• Transitive Property

All three properties should be considered together.

For example, all three properties must be true for the equal (=), congruent ( ) , or similarity ( ) sign to be valid in a mathematical relationship such as a function or an equation.

Reflexive Property: (a=a)

a is equivalent to a
("a" equals "a")

Examples of reflexivity:

2 = 2

x = x

(numbers are listed in
exactly the same order)

5+4 = 5+4

X+2 = X+2

The property of reflexivity may seem ridiculously obvious. However, it is used extensively in proofs.

For example, the property of reflexivity can be used to prove that the following two triangles are congruent. Both triangles have a common side: AC Symmetric Property: (if a=b, then b=a)

The Symmetric and Commutative Property are similar.

a = b is equivalent to b = a

Examples of the Symmetric Property:

(numbers are listed in
reverse order - a reflection)

5+4 = 4+5

X+2 = 2+X

This property is often used to teach elementary school children how numbers work.

For example: Young children are taught that the numbers can be rewritten as follows:  Young children are taught that the numbers can be rewritten as follows: 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.

The logic provided by the Transitive Property can be summarized as follows: if one item is equal to a second item, and the second item is equal to a third item, then the first item is also equal to the third item.

For example, if "Kevin is the doctor for my family" and "the doctor for my family is my cousin", then the transitive property shows that "Kevin is my cousin".

An example of how this principle can be applied to mathematics is:

.5 = 1/2

1/2 = 2/4

Therefore,

.5 = 2/4 Free Resources
Reflexive, Symmetric, and Transitive Properties
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