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Proof: If S = T then, by de nition, S and T have precisely the same elements In particular, this means that x S implies x T and also x T implies x S That is, S T and T S Now suppose that both S T and T S Seeking a contradiction, suppose that S = T Then either there is some element of S that is not an element of T or there is some element of T that is not an element of S The rst eventuality contradicts S T and the second eventuality contradicts T S We conclude that S = T De nition 33 We let denote the set that contains no elements That is, x, x We call the empty set EXAMPLE 34 If S is any set then S To see this, notice that the statement if x then x S must be true because the hypothesis x is false (Check the truth table for if-then statements) This veri es that S EXAMPLE 35 Let S = {x N : x + 2 19 and x < 3} Then S is a sensible set There are no internal contradictions in its de nition But S = There are no elements in S De nition 34 Let S and T be sets We say that x is an element of S T if both x S and x T We say that x is an element of S T if either x S or x T We call S T the intersection of the sets S and T We call S T the union of the sets S and T EXAMPLE 36 Let S = {x N : 2 < x < 9} and T = {x N : 5 x < 14} Then S T = {x N : 5 x < 9}, for these are the points common to both sets And S T = {x N : 2 < x < 14}, for these are the points that are either in S or in T or in both Remark 31 Observe that the use of or in the de nition of set union justi es our decision to use the inclusive or rather than the exclusive or in mathematics See also Proposition 32 below EXAMPLE 37 Let S = {x N : 1 x 5} and T = {x N : 8 < x 12} Then S T = , for the sets S and T have no elements in common On the other hand, S T = {x N : 1 x 5 or 8 < x 12} De nition 35 Let S and T be sets We say that x S \T if both x S and x T We call S\T the set-theoretic difference of S and T
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EXAMPLE 38 Let S = {x N : 2 < x < 7} and T = {x N : 5 x < 10} Then we see that S\T = {x N : 2 < x < 5} and T\S = {x N : 7 x < 10} De nition 36 Suppose that we are studying subsets of a xed set X If S X then we use the symbol c S to denote X\S In this context we sometimes refer to X as the universal set We call c S the complement of S (in X )
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EXAMPLE 39 Let N be the universal set Let S = {x N : 3 < x 20} Then
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S = {x N : 1 x 3} {x N : 20 < x}
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The next proposition puts our use of the inclusive or into context Proposition 32 Let X be the universal set and S X, T X Then (1) c (S T ) = c S c T (2) c (S T ) = c S c T Proof: We shall present this proof in detail since it is a good exercise in understanding both our de nitions and our method of proof We begin with the proof of Part (1) It is often best to treat the proof of the equality of two sets as two separate proofs of containment (This is why Proposition 31 is important) That is what we now do Let x c (S T ) Then, by de nition, x (S T ) Thus x is neither an element of S nor an element of T So both x c S and x c T Hence x c S c T We conclude that c (S T ) c S c T Conversely, if x c S c T then x S and x T Therefore x (S T ) As a result, x c (S T ) Thus c S c T c (S T ) Summarizing, we have c (S T ) = c S c T The proof of Part (2) is similar, but we include it for practice Let x c (S T ) Then, by de nition, x (S T ) Thus x is not both an element of S and an element of T So either x c S or x c T Hence x c S c T We conclude that c (S T ) c S c T Conversely, if x c S c T then either x S or x T Therefore x (S T ) As a result, x c (S T ) Thus c S c T c (S T ) Summarizing, we have c (S T ) = c S c T
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