Basic Set Theory

DEF: For our purposes, a set is any collection of entities. Sometimes we list the elements in a set: {John, Bill, Mary} {5, 77, 12, 1818} {1, 3, 5, 7, 9, . . . } Sometimes we describe the set by using some kind of property: {x: x is an even number} = the set of even numbers {x: x is a number} = the set of all the numbers (and only the numbers). DEF: x ! A iff x is one of the elements in A. Fred ! {Fred, Tom} 7 ! {x: x is a number} 7 ! {x: x is an even number} The empty set (written ! or {} or ") can be defined: (#x)x ! ! DEF: For any sets A and B, A is a subset of B (written A $ B) iff everything that is in A is also in B: (#x)(x A % x ! B) Thus, for any set A, A $ A {7, 6, 8} $ {3, 4, 8, 55, 7, 6, 4} {Ronald Regan, Jimmy Carter} $ { x: x is a former President} {x: x is prime} $ {x: x is a number}