What are the axioms of real numbers and Properties of Equivalence? Examples please. Thank you!
The Real numbers axioms are called:
Real numbers are combined using the basic laws of computation: addition and multiplication
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).
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).
The real set of numbers R contains at least two distinct elements.
Basic axioms of real numbers are: Field Axiom and Extended Axiom.