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{"ops":[{"attributes":{"bold":true},"insert":"2. Prove that a relation R on a set A is symmetric if R"},{"attributes":{"bold":true,"script":"super"},"insert":"-1"},{"attributes":{"bold":true},"insert":" = R."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"3. Give an example of a relation that is reflexive but neither symmetric nor transitive."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"4. Show that the relation \u2018is perpendicular to\u2019 over the set of all straight lines in the plane is symmetric but neither reflexive nor transitive."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"5. Let S and T be sets with "},{"attributes":{"italic":true,"bold":true},"insert":"m "},{"attributes":{"bold":true},"insert":"and "},{"attributes":{"italic":true,"bold":true},"insert":"n "},{"attributes":{"bold":true},"insert":"element respectively. How many elements has S "},{"insert":"\u00d7 "},{"attributes":{"bold":true},"insert":"T? How many relations are there in S "},{"insert":"\u00d7 "},{"attributes":{"bold":true},"insert":"T?"},{"insert":"\n"},{"attributes":{"bold":true},"insert":"6. If R and S are equivalence relations in the set X, prove that R "},{"insert":"\u2229 "},{"attributes":{"bold":true},"insert":"S is an equivalence relation."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"7. Show that the relation of congruence modulo "},{"attributes":{"italic":true,"bold":true},"insert":"m "},{"attributes":{"bold":true},"insert":"has "},{"attributes":{"italic":true,"bold":true},"insert":"m "},{"attributes":{"bold":true},"insert":"distinct equivalence classes."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"8. Show that a partition of a set S deter- mines an equivalence relation in S."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"9. Let S = {"},{"attributes":{"italic":true,"bold":true},"insert":"n"},{"attributes":{"bold":true},"insert":": "},{"attributes":{"italic":true,"bold":true},"insert":"n "},{"insert":"\u2208 "},{"attributes":{"bold":true},"insert":"N and "},{"attributes":{"italic":true,"bold":true},"insert":"n "},{"insert":"> "},{"attributes":{"bold":true},"insert":"1}. If "},{"attributes":{"italic":true,"bold":true},"insert":"a"},{"attributes":{"bold":true},"insert":", "},{"attributes":{"italic":true,"bold":true},"insert":"b "},{"insert":"\u2208 "},{"attributes":{"bold":true},"insert":"S define "},{"attributes":{"italic":true,"bold":true},"insert":"a "},{"attributes":{"bold":true},"insert":"~ "},{"attributes":{"italic":true,"bold":true},"insert":"b "},{"attributes":{"bold":true},"insert":"to mean that "},{"attributes":{"italic":true,"bold":true},"insert":"a "},{"attributes":{"bold":true},"insert":"and "},{"attributes":{"italic":true,"bold":true},"insert":"b "},{"attributes":{"bold":true},"insert":"have the same number of positive prime factors (distinct or identical). Show that ~ is an equivalence relation."},{"insert":"\n"},{"attributes":{"bold":true},"insert":"10. Prove that in the set N "},{"insert":"\u00d7 "},{"attributes":{"bold":true},"insert":"N, the relation R defined by ("},{"attributes":{"italic":true,"bold":true},"insert":"a"},{"attributes":{"bold":true},"insert":", "},{"attributes":{"italic":true,"bold":true},"insert":"b"},{"attributes":{"bold":true},"insert":") R ("},{"attributes":{"italic":true,"bold":true},"insert":"c"},{"attributes":{"bold":true},"insert":", "},{"attributes":{"italic":true,"bold":true},"insert":"d "},{"attributes":{"bold":true},"insert":") "},{"insert":"\u21d4 "},{"attributes":{"italic":true,"bold":true},"insert":"ad "},{"attributes":{"bold":true},"insert":"= "},{"attributes":{"italic":true,"bold":true},"insert":"bc "},{"attributes":{"bold":true},"insert":"is an equivalence relation."},{"insert":"\n\n\n"}]}
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