Lecture 7 21 127 Concepts of Math 09 12 2012 Power Sets continued Another example Use A 1 Then P A 1 1 It is true in this example that A A P A P A What does P N look like Let s write down some examples N P N P N 1 2 3 P N How does this set compare to P Z How about P R It helps to think about these standard objects of mathematics Practicing abstraction and abstract thinking is very helpful Definition For every n N we let n denote the set n x N 1 x n This is convenient notation to clean up summation notation X n n 1 k 2 k n Suppose m n N and m n Observe that m n Does this work the other way Yes When can we say m n We need to know m n and m n Oh look that s the same as m n and m n neat How about P n How many elements does it have How do P m and P n compare In general we can prove the following claim Fact Suppose that A B Then P A P B Proof Suppose that A and B are sets and A B We want to show that P A P B To do that we need to satisfy the formal definition of subset That is we need to show that every element X P A also satisfies X P B To prove this let s take an arbitrary and fixed element X P A We need to explain why X P B as well By definition we know that X P A means X A We assumed that A B Thus X A B We claim that this shows X B To see why we would need to show that whenever we have an element x X it is also true that x B Again this is satisfying the formal definition of If we have an element x X then the fact that X A tells us by definition that x A as well Then the fact that A B tells us by definition that x B as well Thus X B We now know that X B so by definition of power set we deduce that X P B Overall we have shown that every element X P A also satisfies X P B This shows by definition that P A P B 1 Set Operations Definition The intersection of A and B is the set of elements that belong to both A and B and is denoted by A B Symbolically we define A B x U x A and x B Definition The union of A and B is the set of elements that belong to either A or B and is denoted by A B Symbolically we define A B x U x A or x B Remember that or is the inclusive or in mathematics so it doesn t matter if both parts are satisfied That is if x A and x B both then we still say x A B Definition The difference between A and B denoted by A B is the set of elements that belong to A but not to B Symbolically we define A B x U x A and x B Definition The complement of A is the set of all elements not in A and is denoted by A Symbolically we define A x U x A Note it is essential to identify the universal set Definition The Cartesian Product of A and B is written as A B and defined to be A B a b a A and b B This is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B Examples Let A 3 1 2 3 and B and C x 3 8 Then A C 3 B C A B We say A and B are disjoint A B 1 2 3 A C 1 2 3 x 8 A C 1 2 C A x 8 A B A A 5 Note This happens because 3 5 A B 1 1 2 2 3 3 B A A For the last one the context is important If U 5 then A 4 5 If U N then A 4 5 6 Notice that A B and B A are different here Is this always true Fact A and A 2 Fact If A B then A B B A since both sets are just A A Facts observations A B B A and A B B A But in general we can t necessarily say whether A B B A or A B B A Think about when equality does hold and when it doesn t Can you make a claim and prove it Facts A B C A B C A B C A B C A B C You figure this out Higher order Cartesian Products We could also define ordered triples N3 a b c a b c N Notice how this is different from N N N A typical element of N3 looks like say 1 2 3 whereas a typical element of N N N looks like say 1 2 3 There is a natural relationship between the two sets but mathematically speaking they are different objects with different structure Indexed Operations Rather than write out a bunch of similar looking sets we can instead define all of them at once using an index set Define A1 1 0 1 A2 2 0 2 A3 3 0 3 Instead I could say Let I 3 and define for every i 3 Ai i 0 i This is much shorter especially if we were defining say 5 sets or 100 We can then use this same notation to express a big union or intersection of all the sets Ai A1 A2 A3 i 3 Notice that this is okay because as we saw above is associative so it doesn t matter in what order we apply the union operations at the end of the day we will end up with one set whose elements are all of the elements of the Ai sets Here we have Ai 3 2 1 0 1 2 3 i 3 Also notice that Ai 0 i 3 If we changed I to 10 what would happen What if I N 3
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