The power set of a set A is the set of all its subsets including, of course, itself and the empty set. It is denoted by P A.
Look at the cardinality of the four sets in Example 4 , and the cardinality of their corresponding power sets. They are:. Clearly, there's a simple rule at work here: expressed as powers of 2, the cardinalities of the power sets are 2 3 , 2 2 , 2 1 and 2 0.
But can we see why? Well, the elements of the power set of A are all the possible subsets of A. Any one of these subsets can be formed in the following way:. Now the fundamental principle of combinatorics tells us that if we can do each of k things in 2 ways, then the total number of ways of doing them all, one after the other, is 2 k. Each one of these 2 k combinations of decisions - including elements or omitting them - gives us a different subset of A. There are therefore 2 k different subsets altogether.
The laws listed below can be described as the Foundational Rules of Set Theory. We derive them by going back to the definitions of intersection , union , universal set and empty set , and by considering whether a given element is in, or not in, one or more sets. This law might be familiar to you if you've studied logic. The above relationship is comparable to the tautology. These look pretty obvious, don't they? A simple explanation for the intersection part of these laws goes something like this:.
The intersection of two sets A and B is defined as just those elements that are in A and in B. If we replace B by A in this definition we get that the intersection of A and A is the set comprising just those elements that are in A and in A. Obviously, we don't need to say this twice I heard you the first time , so we can simply say that the intersection of A and A is the set of elements in A.
In other words:.
One of these laws looks like this:. If you've done Exercise 3 , question 4, you may have spotted this law already from the Venn Diagrams. Look closely at how this Law works.
Set Theory :DeMorgan's law : Written Proof (Part 1)
The complement symbol after the bracket affects all three symbols inside the bracket when the brackets are removed:. To prove this law, note first of all that when we defined a subset we said that if. We don't know anything about x ; it could be a number, a function, or indeed an elephant. All we do know about x , is that.
This means that x answers No to both questions Are you in A? So by definition. The 'partner laws' in each pair are called duals , each law being the dual of the other. This is called the Principle of Duality. In practical terms, it means that you only need to remember half of this table!
This set of laws constitutes the axioms of a Boolean Algebra. See Boolean Algebra for more.
We may use these laws - and only these laws - to determine whether other statements about the relationships between sets are true or false.
Venn diagrams may be helpful in suggesting such relationships, but only a proof based on these laws will be accepted by mathematicians as rigorous. State carefully which Law you are using at each stage.
A statement like this — one that is true for all values of A and B — is sometimes called an identity. There are no foolproof methods with these proofs — practice is the best guide.
But here are a few general hints. But you can go round in circles with this one.
Try it first before reading the solution below, if you like. Click link for Set Theory Exercise 4. This is sometimes called the r-s Principle. You're getting ready to go out. You have 5 different clean!
basic discrete mathematics logic set theory and probability
In how many ways can you choose to put on a pair of pants and some jeans? In each of the two situations above, we have examples of ordered pairs. As the name says, an ordered pair is simply a pair of 'things' arranged in a certain order.
Often the order in which the things are chosen is arbitrary — we simply have to decide on a convention, and then stick to it; sometimes the order really only works one way round.
Discrete mathematics set theory proofs pdf
In the case of the meals, most people choose to eat their main course first, and then dessert. In the clothes we wear, we put on our underpants before our jeans. You are perfectly free to fly in the face of convention and have your dessert before the main course - or to wear your underwear on top of your trousers - but you'll end up with different sets of ordered pairs if you do.
And, of course, you'll usually have a lot of explaining to do! The two 'things' that make up an ordered pair are written in round brackets, and separated by a comma; like this:.
You will have met ordered pairs before if you've done coordinate geometry. For example, 2, 3 represents the point 2 units along the x -axis and 3 units up the y -axis.
Here again, there's a convention at work in the order of the numbers: we use alphabetical order and put the x -coordinate before the y -coordinate. Again, you could choose to do your coordinate geometry the other way round, and put y before x , but once again, you'd have a lot of explaining to do!
Discrete Mathematics/Set theory/Page 2
C is called the set product or Cartesian product of M and D , and we write:. Suppose that the menu in Example 7 is expanded to include a starter, which is either soup or fruit juice.
How many complete meals can be chosen now? An element of this set is an ordered triple : starter, main course, dessert.
Discrete Mathematics/Set theory/Exercises
In general, if we have n sets: A 1 , A 2 , You probably know already the way in which a point in a plane may be represented by an ordered pair. The diagram in Fig. The lines are called axes : the x -axis and the y -axis. The point where they meet is called the origin.
The point 5, 2 is then located as follows: start at the origin; move 5 units in the direction of the x -axis, and then 2 units in the direction of the y -axis. Example 9. Like this:. The same can also be said about B and Q : B must lie on the x -axis, and Q on the y -axis. Click link for Set Theory Exercise 5. From Wikibooks, open books for an open world. Category : Book:Discrete Mathematics.
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