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Abelian group Abelian semigroup abstract algebra addition and multiplication algebra associative law Cartesian product chapter commutative group Complete the proof complex numbers congruent define definition distributive law element equal equation equivalence relation equivalent ordered pairs example Exercises forms a commutative geometry given Hence homomorphism hypothesis identity integral domain isomorphic line segments linear ordering lower bound mathematical induction mathematicians matrix multiplicative inverse natural numbers negative cut negative integer negative rational numbers number field number q number system numbers forms numbers x one-to-one correspondence pairs of natural plane points polynomials positive cuts positive integer positive rational number postulates proof is left proof of Theorem properties of addition Prove Theorem quaternions reader real numbers representative result ring satisfied set of integers set of natural set of numbers set of rational solution successor Suppose Theorem 1-9 totally ordered trichotomy law vectors verify zero