Natural numbers

Use any properties of the natural numbers to argue that if m and n are natural numbers, then if n x m = m, then n = 1

Well-Ordering of the System of Natural Numbers

Prove that every non-empty set of natural numbers contains a least natural number. (Hint: Let A be a set of natural numbers that does not contain a least natural number and let S be the set of natural numbers not in set A. Then argue that set S contains all natural numbers so tha set A must be empty).

Natural numbers

Use any properties of the natural numbers, N, to argue that is S = N x N and R is the relation on R defined by (x,y)R(u,v) means x + v = y + u, then R is equivalence relation on S

Integer proof

Prove Theorem 31 and definition 24 of the attached file If (c,d)^R is an integer...

Integer proof

Prove theorem 32 (see attached)... If (a,b)^R and (c,d)^R are integers....

Integer proof

prove theorem 33 and theorem 34 of the attached file If (a,b)^R, (c,d)^R, and (e,f)^R are integers....

Multiplicative identity proof

Prove definition 26 in attachment The integer (a+1,a)^R has an important property with respect to multiplication.....

Integer proof

Prove If (c,d)^R is any integer, then (a + 1,a)^R x (c,d)^R = (c,d)^R

Multiplicative identity proof

Prove Theorem 36 of the attached file Theorem 36 The multiplicative identity for the set of integers is unique

Equivalence class proof

Prove Theorem 37 of the attached file For integers (a,b)^R,(c,d)^R and (e,f)^R, (a,b)^R x ((c,d)^R+(e,f)^R)=(a,b)^R x (c,d)^R + (a,b)^R x (e,f)^R