4 SET THEORY [CHAP.1 However,if A and B are two arbitrary sets,it is possible that some objects are in A but not in B,some are in B but not in A,some are in both A and B,and some are in neither A nor B;hence in general we represent A and B as in Fig.1-1(c). Arguments and Venn Diagrams Many verbal statements are essentially statements about sets and can therefore be described by Venn diagrams. Hence Venn diagrams can sometimes be used to determine whether or not an argument is valid. EXAMPLE 1.3 Show that the following argument (adapted from a book on logic by Lewis Carroll,the author of Alice in Wonderland)is valid: S:All my tin objects are saucepans. S2:I find all your presents very useful. S3:None of my saucepans is of the slightest use. S:Your presents to me are not made of tin. The statements S1,S2,and S3 above the horizontal line denote the assumptions,and the statement S below the line denotes the conclusion.The argument is valid if the conclusion S follows logically from the assumptions S1,S2,and S3. By SI the tin objects are contained in the set of saucepans,and by S3 the set of saucepans and the set of useful things are disjoint.Furthermore,by S2 the set of"your presents"is a subset of the set of useful things. Accordingly,we can draw the Venn diagram in Fig.1-2. The conclusion is clearly valid by the Venn diagram because the set of"your presents"is disjoint from the set of tin objects. tin objects your presents sauscpans useful things Fig.1-2 1.4 SET OPERATIONS This section introduces a number of set operations,including the basic operations of union,intersection,and complement. Union and Intersection The union of two sets A and B,denoted by A U B,is the set of all elements which belong to A or to B; that is, AUB={xIx∈Aorx∈B} Here"or"is used in the sense of and/or.Figure 1-3(a)is a Venn diagram in which AU B is shaded. The intersection of two sets A and B,denoted by An B,is the set of elements which belong to both A and B:that is, AnB={xlx∈A andx∈B) Figure 1-3(b)is a Venn diagram in which An B is shaded.4 SET THEORY [CHAP. 1 However, if A and B are two arbitrary sets, it is possible that some objects are in A but not in B, some are in B but not in A, some are in both A and B, and some are in neither A nor B; hence in general we represent A and B as in Fig. 1-1(c). Arguments and Venn Diagrams Many verbal statements are essentially statements about sets and can therefore be described by Venn diagrams. Hence Venn diagrams can sometimes be used to determine whether or not an argument is valid. EXAMPLE 1.3 Show that the following argument (adapted from a book on logic by Lewis Carroll, the author of Alice in Wonderland) is valid: S1: All my tin objects are saucepans. S2: I find all your presents very useful. S3: None of my saucepans is of the slightest use. S : Your presents to me are not made of tin. The statements S1, S2, and S3 above the horizontal line denote the assumptions, and the statement S below the line denotes the conclusion. The argument is valid if the conclusion S follows logically from the assumptions S1, S2, and S3. By S1 the tin objects are contained in the set of saucepans, and by S3 the set of saucepans and the set of useful things are disjoint. Furthermore, by S2 the set of “your presents” is a subset of the set of useful things. Accordingly, we can draw the Venn diagram in Fig. 1-2. The conclusion is clearly valid by the Venn diagram because the set of “your presents” is disjoint from the set of tin objects. Fig. 1-2 1.4 SET OPERATIONS This section introduces a number of set operations, including the basic operations of union, intersection, and complement. Union and Intersection The union of two sets A and B, denoted by A ∪ B, is the set of all elements which belong to A or to B; that is, A ∪ B = {x | x ∈ A or x ∈ B} Here “or” is used in the sense of and/or. Figure 1-3(a) is a Venn diagram in which A ∪ B is shaded. The intersection of two sets A and B, denoted by A ∩ B, is the set of elements which belong to both A and B; that is, A ∩ B = {x | x ∈ A and x ∈ B} Figure 1-3(b) is a Venn diagram in which A ∩ B is shaded