The twelve essential properties of the set of real numbers are defined here as axioms:
(P1) (Associative law for addition)
$a + (b + c) = (a + b) + c$
(P2) (Existence of an additive identity)
$a + 0 = 0 + a = a$
(P3) (Existence of additive inverses)
$a + (-a) = (-a) + a = 0$
(P4) (Commutative law for addition)
$a + b = b + a$
(P5) (Associative law for multiplication)
$a \cdot (b \cdot c) = (a \cdot b) \cdot c$
(P6) (Existence of a multiplicative identity)
$a \cdot 1 = 1 \cdot a = a$
(P7) (Existence of multiplicative inverses)
$a \cdot a^{-1} = a^{-1} \cdot a = 1, \quad \text{for} \quad a \neq 0$
(P8) (Commutative law for multiplication)
$a \cdot b = b \cdot a$
(P9) (Distributive law)
$a \cdot (b+c) = a \cdot b + a \cdot c$
All numbers $a$ that satisfy $a>0$ will be defined to be positive while all numbers $a$ that satisfy $a<0$ are defined as negative. For convenience, we will consider the set of all positive numbers, denoted by $P$, and state all remaining properties in terms of $P$:
(P10) (Trichotomy law)
For every number $a$, one and only one of the following holds:
(i) $a = 0$
(ii) $a$ is in the collection $P$,
(iii) $-a$ is in the collection $P$(P11) (Closure under addition)
If $a$ and $b$ are in $P$, then $a + b$ is in $P$.
(P12) (Closure under multiplication)
If $a$ and $b$ are in $P$, then $a \cdot b$ is in $P$.
Problems:
| Question Number | Difficulty | Completed |
|---|---|---|
| 1 | () | No |
| 2 | () | No |
| 3 | () | No |
| 4 | () | No |
| 5 | () | No |
| 6 | () | No |
| 7 | () | No |
| 8 | () | No |
| 9 | () | No |
| 10 | () | No |
| 11 | () | No |
| 12 | () | No |
| 13 | () | No |
| 14 | () | No |
| 15 | (*) | No |
| 16 | (*) | No |
| 17 | () | No |
| 18 | () | No |
| 19 | () | No |
| 20 | () | No |
| 21 | (*) | No |
| 22 | (*) | No |
| 23 | (*) | No |
| 24 | (*) | No |
| 25 | () | No |
