View by Category

Loading...

PPT – The Empty Set, Partitions, and Power Sets PowerPoint presentation | free to view - id: e5384-ZDM0M

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Section 5.3

- The Empty Set, Partitions, and Power Sets

Element Method for Proving a Set Is the Empty Set

- To prove that a set X is equal to the empty set

?, - Suppose that X has at least one element.
- Derive a contradiction.

Theorem For any set A, A ? AC ?.

- Proof By contradiction. Suppose there is a set

A such that A ? AC ? ?. Then there is an element

x such that x ? A ? AC. By the definition of

intersection, this means that x ? A and x ? AC.

By the definition of complement, we know that x ?

A and x ? A. This is impossible, and so is a

contradiction. Therefore, for any set A, A ? AC

?. ?

Some Properties Involving ?

- For all sets A, A ? ? A.
- For all sets A, A ? AC ? and A ? AC U.
- For all sets A, A ? ? ?.
- Finally, UC ? and ?C U.

Disjoint Sets

- Two sets are called disjoint if and only if they

have no elements in common. - Symbolically A and B are disjoint ? A ? B ?.

Theorem For all sets A and B, the sets A B and

A ? B are disjoint.

- Proof By contradiction. Assume there are sets A

and B such that A B and A ? B are not disjoint.

Then (A B) ? (A ? B) has at least one

element. Let x be an element of (A B) ? (A ?

B). By the definition of intersection, this

means that x ? A B and x ? A ? B. Since x ? A

B, x is in A but not in B, and since x ? A ? B,

x is in both A and B. This is a contradiction

because x cannot both be in B and not in B.

Therefore, no such element x exists, so A B and

A ? B must be disjoint. ?

Mutually Disjoint Sets

- Sets A1, A2, , An are said to be mutually

disjoint (or pairwise disjoint or nonoverlapping)

if and only if no two sets Ai and Aj with

distinct (different) subscripts have any elements

in common. So.for all i, j 1, 2, , n, Ai ?

Aj ? whenever i ? j. - Example The sets 1, 2, 3, 5, and 4, 7 are

mutually disjoint.

Partitions

- A collection of nonempty sets A1, A2, , An is

a partition of a set A if and only if - A A1 ? A2 ? ? An
- A1, A2, , An are mutually disjoint.
- Example Let A 1, 2, 3, 4, 5, 6. Then B

1, 2, C 3, 5, and D 4, 6 is a partition

of A.

Power Sets

- Given a set A, the power set of A, denoted P(A),

is the set of all subsets of A. - Example If A a, b, c, then
- P(A) ?, a, b, c, a, b, a, c, b, c,

a, b, c

Theorem For all sets A and B, if A ? B, then

P(A) ? P(B).

- Proof Suppose A and B are sets such that A ? B.

Suppose X? P(A). We must show that X? P(B).

Since X? P(A), we have X ? A by the definition of

power sets. But A ? B, so X ? B by the

transitive property of subsets. Therefore, X?

P(B) by the definition of power sets. Since any

element of P(A) is an element of P(B), we have

P(A) ? P(B) by the definition of subsets. ?

Theorem for all integers n ? 0, if a set X has n

elements then P(X) has 2n elements.

- Outline of proof
- Well use induction on the number of elements in

X. - We can restate the theorem as Any set with n

elements has 2n subsets. - True for n 0 P(?) ?.
- Assume our restatement is true for n k.

Theorem for all integers n ? 0, if a set X has n

elements then P(X) has 2n elements.

- Consider a set X with k 1 elements, and let a

be an element of X. - Every subset of X either contains a or doesnt

contain a. - X - a has k elements, so the number of subsets

of X without a in them is 2k by our inductive

hypothesis. - (Tricky part) Subset of X that do contain a can

be matched exactly with the subsets of X that do

contain a. - Therefore, X has 2(2k) 2k1 subsets.

Fridays class

- Start section 7.1.

Information About the Test

- The average was about 80.
- The high score was 100.
- The low score was 42.5.

About PowerShow.com

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "The Empty Set, Partitions, and Power Sets" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Gatech University and other schools with their online training by sharing educational presentations for free