History of Basic Set Theory

	Revision 41 . . December 7, 2001 11:55 pm by The Anome [more typesetting...]
	Revision 40 . . December 7, 2001 11:24 pm by The Anome [linked 'axiom']
	Revision 39 . . December 7, 2001 11:22 pm by The Anome [reformatting, hopefully without breaking anything]
	Revision 38 . . (edit) September 30, 2001 9:22 pm by Zundark [link]

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Difference (from prior major revision) (author diff)

Changed: 10,11c10,11

:(i) Order of elements is immaterial. Thus {1,2} = {2,1} :(ii) Repetition of elements is irrelevent, so that {1,2,2,1,1} = {1,2}

:(i) Order of elements is immaterial. Thus {1,2} = {2,1} :(ii) Repetition of elements is irrelevent, so that {1,2,2,1,1} = {1,2}

Changed: 13,22c13,23

We use the notation {x : P} to denote the set containing all objects x such that the condition P holds. For example, : {x : x is real} : denotes the set of real numbers, : {x : x has blonde hair} : denotes the set of all objects which have blonde hair, and : {x : x is a dog} : denotes the set of all dogs. In informal contexts we might also denote this last set by {dogs}. :Sets are frequently named by upper-case letters. Thus we might write :A = {x: x is an even integer} :to mean that A is the set of even integers.

We use the notation {x : P} to denote the set containing all objects x such that the condition P holds. For example, : {x : x is real} denotes the set of real numbers, : {x : x has blonde hair} denotes the set of all objects which have blonde hair, and : {x : x is a dog} denotes the set of all dogs. In informal contexts we might also denote this last set by {dogs}. Sets are frequently named by upper-case letters. Thus we might write :A = {x: x is an even integer} to mean that A is the set of even integers.

Changed: 26,29c27,30

:A = {x : x is real} :B = {x : x is an integer} :C = {x : x is an odd integer} :D = {x : x is or was a US President}

:A = {x : x is real} :B = {x : x is an integer} :C = {x : x is an odd integer} :D = {x : x is or was a US President}

Changed: 38c39

:{x in X : P(x)}

:{x in X : P(x)}

Changed: 40c41

:{x : x is in X and P(x)}

:{x : x is in X and P(x)}

Changed: 44c45

:A' = {x in U: x is not in A}

:A' = {x in U: x is not in A}

Changed: 46c47

:C' = {x in B : x is not an odd integer} = {x : x is an even integer}

:C' = {x in B : x is not an odd integer} = {x : x is an even integer}

Changed: 48c49

:B' = {x : x is real and not an integer}

:B' = {x : x is real and not an integer}

Changed: 51,52c52,53

:A \/ B = {x : x is in A or x is in B or both} :A /\ B = {x : x is in A and x is in B} = {x in A : x is in B} = {x in B : x is in A}

:A \/ B = {x : x is in A or x is in B or both} :A /\ B = {x : x is in A and x is in B} = {x in A : x is in B} = {x in B : x is in A}

Changed: 54,55c55,56

:A = {x in U : x is left handed} :B = {x in U : x has blonde hair}

:A = {x in U : x is left handed} :B = {x in U : x has blonde hair}

Changed: 59,60c60,61

:E = {x : x is a human being} :F = {x : x is over 200 years old}

:E = {x : x is a human being} :F = {x : x is over 200 years old}

Changed: 67,76c68,77

:(a) A /\ B = B /\ A :(b) A \/ B = B \/ A :(c) ( A /\ B) /\ C = A /\ ( B /\ C) :(d) ( A \/ B) \/ C = A \/ ( B \/ C) :(e) A is a subset of B if and only if A \/ B = B :(f) A is a subset of B if and only if A /\ B = A :(g) A /\ B is a subset of A, which is a subset of A \/ B :(h) A /\ O = O :(i) A \/ O = A :(j) If A and B are subsets of a universal set U, then A is a subset of B if and only if B' is a subset of A'

:(a) A /\ B = B /\ A :(b) A \/ B = B \/ A :(c) (A /\ B) /\ C = A /\ (B /\ C) :(d) (A \/ B) \/ C = A \/ (B \/ C) :(e) A is a subset of B if and only if A \/ B = B :(f) A is a subset of B if and only if A /\ B = A :(g) A /\ B is a subset of A, which is a subset of A \/ B :(h) A /\ O = O :(i) A \/ O = A :(j) If A and B are subsets of a universal set U, then A is a subset of B if and only if B' is a subset of A'

Changed: 80,81c81,82

:(a) A /\ ( B \/ C) = ( A /\ B) \/ ( A /\ C) :(b) A \/ ( B /\ C) = ( A \/ B) /\ ( A \/ C)

:(a) A /\ ( B \/ C) = (A /\ B) \/ (A /\ C) :(b) A \/ ( B /\ C) = (A \/ B) /\ (A \/ C)

Changed: 83,85c84,87

:(i) Pick any element x of the left-hand side (LHS). Then, by definition of /\, x is in A and x is in B \/ C; that is x is in A and either x is in B or x is in C. In the first case, x is in both A and B, so is in A /\ B and a fortiori in ( A /\ B) \/ ( A /\ C). In the second case x is in both A and C and so is again in ( A /\ B) \/ ( A /\ C). Thus in either case x is in ( A /\ B) \/ ( A /\ C). we have shown that every element of the LHS is automatically in the RHS. But this is precisely what we mean by saying that the LHS is a subset of the RHS. :(ii) Pick any element x of the RHS. Then x is in ( A /\ B) or x is in ( A /\ C); in the first case x is in A and x is in B; in the second, x is in A and x is in C. In either case x is in A. Also x in the first case x is in B and hence in B \/ C; in the second case x is in C and thus in B \/ C. We have proved that whatever x is, if it is a member of the RHS then it is in both A and B \/ C and hence by definition is in A /\ ( B \/ C). We have proved that the RHS is a subset of the LHS.

:(i) Pick any element x of the left-hand side (LHS). Then, by definition of /\, x is in A and x is in B \/ C; that is x is in A and either x is in B or x is in C. In the first case, x is in both A and B, so is in A /\ B and a fortiori in (A /\ B) \/ (A /\ C). In the second case x is in both A and C and so is again in (A /\ B) \/ (A /\ C). Thus in either case x is in (A /\ B) \/ (A /\ C). we have shown that every element of the LHS is automatically in the RHS. But this is precisely what we mean by saying that the LHS is a subset of the RHS. :(ii) Pick any element x of the RHS. Then x is in (A /\ B) or x is in (A /\ C); in the first case x is in A and x is in B; in the second, x is in A and x is in C. In either case x is in A. Also x in the first case x is in B and hence in B \/ C; in the second case x is in C and thus in B \/ C. We have proved that whatever x is, if it is a member of the RHS then it is in both A and B \/ C and hence by definition is in A /\ (B \/ C). We have proved that the RHS is a subset of the LHS.

Changed: 90c92

:Z = {x : x is not a member of itself}

:Z = {x : x is not a member of itself}

Removed: 94,95d95

:A * B = { (a , b) : a is in A and b is in B } That is, A * B is the set of all ordered pairs whose first component is an element of A and whose second component is an element of B.

Changed: 97c97,100

We can extend this definition to a set A * B * C of ordered triples, and more generally to sets of ordered n-tuples for any positive integer n. It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product.

:A * B = { (a , b) : a is in A and b is in B } That is, A * B is the set of all ordered pairs whose first component is an element of A and whose second component is an element of B. We can extend this definition to a set A * B * C of ordered triples, and more generally to sets of ordered n-tuples for any positive integer n. It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product.