{A, B, C} which maps every element onto the equivalence class to which it belongs, thus PR(I) = PR(7) = B. }, and of the arithmetical operations defined upon it: one starting point is the system of axioms due to Peano, the other a construction based upon set theory and the Zermelo-Fraenkel Axioms in which system the Peano Axioms will have the status of theorems.

N = {o, 31 is a normal subgroup of 718 and so we can form the quotient group 718/N which comprises the three cosets A = {o, 3}, B = {i, 41 and C = {2, 51 and which is isomorphic to the group 713 under the isomorphism 0 defined by O(A) = o, c(B) = I, ยข(C) = 2. j. I is an ideal of 71 and the quotient ring 71/I is isomorphic to 718. Note that g, as defined above, is not a homomorphism of rings since it does not map the multiplicative identity element of 71 onto that of 718. See also: Cayley's Theorem (p.

It is defined by 7O = 'N, ao(") = o- o v" (all n e N). ('N denotes the identity function on N, see p. ) Inequalities are defined on N by m is smaller (or less) than n (m < n) if and only if there exists x+ o in N such that n = m+x. [ 20 Number systems / We write 21 m < n for m n for n < m (m is greater than n), m,>n for n, n holds; (b) addition and multiplication are commutative (p.