What are the axioms of real numbers and Properties of Equivalence? Examples please. Thank you!

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The Real numbers axioms are called:

1)Field Axiom

Real numbers are combined using the basic laws of computation: addition and multiplication

Addition laws:

a) The sum of any two real numbers is also a real number.

x in R; y in R => x+y in R

b) The addition operation is commutative in R: x+y = y+x

c) The addition operation is associative in R: (x+y)+z = x+(y+z)

d) Additive identity: 0+x = x+0 = x

e) Additive inverse:

`EE` y in R such that x+y = y+x= 0 => x = -x (opposite or negative of x).

Multiplication laws:

a) The product of any two real numbers is also a real number.

x in R; y in R => x*y in R

b) The multiplication operation is commutative in R: x*y = y*x

c) The multiplication operation is associative in R: (x*y)*z = x*(y*z)

d) Multiplicative identity: 1*x = x*1 = x

e) Multiplicative inverse:

``y in R such that x*y = y*x= 1 => x = 1/x (inverse of x).

2)Extended Axiom;

The real set of numbers R contains at least two distinct elements.

Basic axioms of real numbers are: Field Axiom and Extended Axiom.

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