Abstract Algebra Theory and Applications - Computer Science ...

0.2 SETS AND EQUIVALENCE RELATIONS 5if each x in X satisfies a certain property P. For example, if E is the set ofeven positive integers, we can describe E by writing eitherE = {2, 4, 6, . . .}orE = {x : x is an even integer and x > 0}.We write 2 ∈ E when we want to say that 2 is in the set E, and −3 /∈ E tosay that −3 is not in the set E.Some of the more important sets that we will consider are the following:N = {n : n is a natural number} = {1, 2, 3, . . .};Z = {n : n is an integer} = {. . . , −1, 0, 1, 2, . . .};Q = {r : r is a rational number} = {p/q : p, q ∈ Z where q ≠ 0};R = {x : x is a real number};C = {z : z is a complex number}.We find various relations between sets and can perform operations onsets. A set A is a subset of B, written A ⊂ B or B ⊃ A, if every elementof A is also an element of B. For example,{4, 5, 8} ⊂ {2, 3, 4, 5, 6, 7, 8, 9}andN ⊂ Z ⊂ Q ⊂ R ⊂ C.Trivially, every set is a subset of itself. A set B is a proper subset of aset A if B ⊂ A but B ≠ A. If A is not a subset of B, we write A ⊄ B; forexample, {4, 7, 9} ⊄ {2, 4, 5, 8, 9}. Two sets are equal, written A = B, if wecan show that A ⊂ B and B ⊂ A.It is convenient to have a set with no elements in it. This set is calledthe empty set and is denoted by ∅. Note that the empty set is a subset ofevery set.To construct new sets out of old sets, we can perform certain operations:the union A ∪ B of two sets A and B is defined asA ∪ B = {x : x ∈ A or x ∈ B};the intersection of A and B is defined byA ∩ B = {x : x ∈ A and x ∈ B}.