lecture10(complete)

# lecture10(complete) - MA1100 Lecture 10 Mathematical Proofs...

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MA1100 Lecture 10 Mathematical Proofs Proving involving ± Sets relations ± Power sets ± Cartesian products ± Indexed collection of sets ± Empty sets Chartrand: section 4.4, 4.5, 4.6

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Lecture 10 2 Announcement ± Homework set 2 due next Tuesday ± Homework 1 solutions scores in IVLE ± MA1100 Wiki has been launched http://wiki.nus.edu.sg/display/MA1100/ ± (Temporary) return of clickers next Tuesday
Lecture 10 3 Proving A Œ B Example A = { x œ Z | 6 | x } B = { x œ Z | x is even } Listing A = {. .., -6, 0, 6, 12, . ..} B = {. .., -4, -2, 0, 2, 4, . ..} A Œ B When the set is large or infinite , it is not rigorous to “prove” subset relationship by listing. Prove that A Œ B A, B are subsets of universal set Z By inspection, every element of A is also an element of B ?

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Lecture 10 4 Element-Chasing Method Choose an arbitrary element with the property given in A Show that the element satisfies the property given in B ( " x œ U) [(x œ A) (x œ B)] Start with x œ A End with x œ B Do not choose a specific element from A Arbitrary element means general element Denote the chosen element by some symbol Use only the given property of the element A Œ B every element of A is also an element of B is equivalent to Direct Proof
Lecture 10 5 Proving A Œ B Example A = { x œ Z | 6 | x } B = { x œ Z | x is even } Proof Prove that A Œ B ( " x œ Z ) [(x œ A) (x œ B)] Let x œ A . Then 6 | x , So x = 2(3k). In other words, x is even . This implies x œ B . Hence we have proven A Œ B. i.e. x = 6k for some integer k.

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Lecture 10 6 Proving A Œ B Example A = { x œ Z | 4 | x 2 } B = { x œ Z | 4 | x } Proof Take x = 2 . Then 4 | 2 2 This implies 2 B . Prove that A Œ B ( \$ x œ Z ) [(x œ A) (x B)] i.e. 2 œ A . But 4 | 2 Hence we have proven A Œ B. ~( " x œ U) [(x œ A) (x œ B)] Constructive proof So (2 œ A) and (2 B)
Lecture 10 7 Proving A = B Example A = { x œ Z | 4 | x 2 } B = { x œ Z | 2 | x } Proof Prove that A = B Let x œ A . So x œ B . Hence we have proven A = B. (i) A Œ B (ii) B Œ A : Let x œ B . So x œ A . : This proves A Œ B. This proves B Œ A. A = B if and only if A Œ B and B Œ A ( " x œ U) [(x œ A) (x œ B)] Let x B . So x A .

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Lecture 10 8 Proving A B There is an element x in U such that A B either x is in A but not in B or x is in B but not in A Either A Œ B or B Œ A . ~( A Œ B and B Œ A ) Constructive proof
Lecture 9 9 Set Operations vs Logical Operations Set Meaning Logic A B A » B A c A – B P: x œ A Q: x œ B x œ A and x œ B x œ A or x œ B x A x œ A and x B P Q P ¤ Q ~P P ~Q Many properties of set operations

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lecture10(complete) - MA1100 Lecture 10 Mathematical Proofs...

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