what are the 2 equality axioms of real numbers and the 5 axioms of order? And can you give examples? Thank You!

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We'll begine with reflexive axiom:

a = a, for ay real number "a"

Example 5 = 5

We'll continue with symmetric axiome:

If a = b, then b = a

Example: If 3 + 4 = 7, then 7 = 3 + 4

Another axiom is transitive axiom:

If a = b and b = c, then a = c

Example: Since 3 + 4 = 7, and 1 + 2 = 3, then 1 + 2 + 3 + 4 = 3 + 7 = 10

The axioms of order:

1) Translation invariance of order: x < y => x + z < y + z, for any x and y real numbers

Example: 2 < 3 => 2 + 4 < 3 + 4 <=> 6 < 7

2) Transitivity of order: x < y and y < z => x < z

3) Trichotomy: x < y and y < x => x = y

4) Scaling:

If x < y and z> 0 => x*z < y*z

**The 2 equality axioms of real numbers are as follows:**

**1) Reflexive axiom of equality** which states that a = a, or any real number equals itself. Example: 5=5

**2) Symmetric axiom of equality** which states that if a = b, then b = a. Example: 12/4 = 3, then 3 = 12/4

**There are 4 aximoms of order for real numbers are as follows:**

**1) Axiom of comparison** which states that for real numbers only one of the following relationships can exist: a > b; a < b; or a = b. Some examples are as follows:

If a = 5 and b = 4, then 5 > 4

If a = 4 and b =5, a< b

If a = 5 and b = 5, then a = b.

**2) Transitive axiom of comparison** which states if a < b and b < c, then a < c.

Example: if a = 4; b =5 and c = 6, then the following is true: 4 < 5 and 5< 6, therefore 4 < 6.

**3) Multiplication axiom of comparison** which states the following:

If a < b and c > 0, then ac < bc

Example: if a = 4, b = 5, and c =6, then (4)(5) < (5)(6)

**4) Additive axiom of comparison** which states the following:

If a < b then a + c < b +c

Example: if a = 4, b = 5, and c = 6, then 4 + 6 < 5 + 6

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