• How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?
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• Does it matter what we do first? second? third? . . .
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• Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?
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• Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?
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• Associative Property
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• Commutative Property
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• Distributive Property
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• Identity Property
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• Inverse Property

• Reflexive Property
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• Symmetric Property
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• Transitive Property

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Examples of Reflexive, Symmetric, and Transitive Equivalence Properties

An Equivalence Relationship always satisfies three conditions:

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.

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 .

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.