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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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 ?
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 6
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 .
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ΠB or B ΠA . ~( A ΠB and B ΠA ) Constructive proof
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 32

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online