Assume B is a Boolean Algebra. Prove the following statement
using only the axioms for a Boolean Algebra properties of a Boolean
Uniqueness of 0: There is only one element of B that is an
identity for +
please include all the steps.
Using field axioms and order axioms prove the following theorems
(explain every step by referencing basic axioms)
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions...
Prove the following theorem. Using the ruler function axiom. List
all axioms and definitions used.
Let P and Q be two points, then the line segment AB=BA (AB and
BA have lines over them to show line segments)
Using field axioms, prove the following theorems:
(i) If x and y are non-zero real numbers, then xy does not equal
(ii) Let x and y be real numbers. Prove the following
1. (-1)x = -x
2. (-x)y = -(xy)=x(-y)
3. (-x)(-y) = xy
(iii) Let a and b be real numbers, and x and y be non-zero real
numbers. Then a/x + b/y = (ay +bx)/(xy)
Please justify and prove each statement (Use explicitly the four
a) Prove that a finite positive linear combination of metrics is
a metric (Use explicitly the four axioms). If it is infinite, will
it be metric?
b) Is the difference between two metrics a metric?
(d1 - d2)