FOR THE STUDENT OF MATHEMATICS
PROPERTIES LIST 4.

Starting with Property XIX on Properties List 1, show how each property results from earlier properties.

Assume that p is any real number, q is any real number, r is any real number and t is any real number.

INEQUALITIES       [ Note: replace   ⇔   with   " if and only if ". ]

If p and q locate points on a horizontal line and p is less than q, p < q , then we consider p on the left of q.

Definition 8. p is less than q written p < q   ⇔   there is a positive number r such that p + r = q.

Definition 9. p is greater than q written p > q   ⇔   there is a positive number r such that p = q + r .

Definition 10. p is less than or equal to q written p ≤ q   ⇔   p = q   or   p < q

Definition 11. p is greater than or equal to q written p ≥ q   ⇔   p = q   or   p > q

Definition 11. p is NOT (an INEQUALITY above) q written p (that INEQUALITY with a vertical line through it) q
⇔   a different INEQUALITY above that you can determine correct by negation.

PROPERTY
LVI.     p is less than 0   ⇔   p < 0   ⇔   p is negative.

LII. RESTATED:     | p |   ≥   0

LVII.     | p | = r   ⇔   p = r   or   p = -r   where   r ≥ 0

XVII. RESTATED:     If p < q   and q < r,   then p < r.

LVIII.     If p < q   and r < t,   then p + r < q + t.
* may replace < with any other inequality and + with - within reason.

LIX.     If p < q   and r > 0,   then p r < q r.       ( see * above for general idea )

LX.     If p < q   and r < 0,   then p r > q r.       ( see * above for general idea )

* Properties XVII, LVIII, LIX and LX are prototypes for many other inequality properties.

LXI.     | p - q | = r     ⇔     p = q - r   or   p = q + r

LXII.     | p - q | < r     ⇔     q - r < p < q + r

LXIII.     | p - q | > r     ⇔     p < q - r   or   p > q + r