If You Don’t Know What Natural Numbers Are, Don’t Even Bother Taking the CSET

Feb
18

Filed Under CSET Math |

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Excerpt from the CSET Math Study Guide

Math teacher candidates studying for the CSET math need to know what natural numbers are.

Natural Numbers

The Peano Postulates

Let there exist a non-empty set N such that
Postulate I: 1 Î N
Postulate II: For each n Î N there exists a unique n* Î N, called the successor of n.
Postulate III: For each n Î N we have n* ¹ 1.
Postulate IV: If m, n Î N and m* = n*, then m = n.
Postulate V: Any subset K of N having the properties

(a) 1 Î K
(b) k* Î K whenever k Î K is equal to N.

Postulates I and II need no elaboration; III states that there is a first natural number 1; IV states that distinct natural numbers m and n have distinct successors m + 1 and n + 1;
V states that any natural number can be reached by beginning with 1 and counting consecutive successors.

ADDITION ON N

Addition on N is defined:

(i) n + 1 = n*, for every n Î N
(ii) n + m* = (n + m)* whenever n + m is defined.

Addition is then subject to the following laws:

For all m, n, p Î N,
A1. Closure Law           n + m Î N
A2. Commutative Law   n + m = m + n

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