Subsets Suppose A and B are sets If every element of A is also an element of B then we say A is a subset of B and we write A B We write A cid 54 B if A is not a subset of B Examples 1 0 4 6 0 3 4 5 6 9 2 0 4 6 cid 54 0 3 4 5 3 0 4 6 0 4 6 Some Observations 1 A for every set A 2 A A for every set A 3 4 N Z Q R Subsets vs Elements Do not confuse subsets and elements 1 2 1 2 3 4 but 2 cid 54 1 2 3 4 2 2 cid 54 1 2 3 4 but 2 1 2 3 4 This gets a little confusing when considering sets that are elements of other sets Let A 1 1 1 1 A 2 1 A and 1 A 3 1 A Subsets What are all the subsets of the set A a b c Set of all subsets of A a b c a b a c b c a b c Powerset The power set of a set A denoted P A is the set of all subsets of A Examples 1 P 2 P a a 3 P a b c a b c a b a c b c a b c Observe that if A is a nite set of cardinality n then P A has 2n elements In other words A has 2n di erent subsets Subsets of R2 Many subsets of R2 can be visualized as regions or curves on the plane Examples 1 C x y R2 x 2 y 2 1 is the unit circle centered at the origin 2 S x y R2 y x 2 x x 2 x R is the curve of the parabola y x 2 3 Q1 x y R2 x y 0 is the rst quadrant Exercises Exercises 1 Explicitly write down the following sets by listing their elements between braces 1 1 The power set of 1 2 3 4 1 2 The power set of 1 1 1 1 1 3 The power set of N Q 1 4 P P 1 1 5 The set X P a b c d X 2 2 Determine the cardinality of the following sets 2 1 The power set of 1 2 3 4 2 2 The power set of 1 2 3 4 a b c 2 3 1 2 3 4 P a b c 2 4 P P 1 2 3 2 5 The set X a b c d X 1
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