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The symbol is also used when we want to read from right to left: B A means A B. Proposition 5. Let A,B,C be sets. We need to show that if x A then x C. Given x A, A B implies that x B. Since B C, this implies that x C. To prove that two sets are equal often involves hard work we have to establish the two subset relations in 5. Sometimes the same set can be described in two apparently different ways. This proposition might be too simple to be interesting. We have included it to illustrate how one proves that two sets are equal.


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We must prove that A B and B A. The first statement means x A x B. So let x A. This proves A B. This proves B A and establishes our desired set equality. Prove that A B and B A.

Project 5. Determine which of the following set equalities are true.

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If a statement is true, prove it. If it is false, explain why this set equality does not hold. The sets A and F will make an appearance in Project Here are some facts about equality of sets: Proposition 5.

These three properties should look familiar we mentioned them already in Section 1. We called the properties reflexivity, symmetry, and transitivity, respectively. When reading or writing a set definition, pay attention to what is a variable inside the set definition and what is not a variable. As examples, how do the following pairs of sets differ? We will see these properties again in Section 6.

The subscripts on T m,v m,w m are not necessary, but this notation is often useful to emphasize the fact that m is a constant. Find the simplest possible way of writing each of these sets. The empty set, denoted by, has the feature that x is never true. We allow ourselves to say the empty set because there is only one set with this property: Proposition 5. The existence of is one of the hidden assumptions mentioned in Section 1.

Assume that the sets 1 and 2 have the property that x 1 is never true and x 2 is never true. We first consider the case 1 2. But that cannot be, since there is no x 1. The other case, 1 2, is dealt with similarly. The empty set is a subset of every set, that is, for every set S, S. Read through the proof of Proposition 5. Then there exists no x that is in A.

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Do you see why the proof still holds? The set operations and give us alternative ways of writing certain sets. Example 5. Project This is a continuation of Project 5. Again, determine which of the following set equalities are true. Project Determine which of the following statements are true for all sets A, B, and C. If a double implication fails, determine whether one or the other of the possible implications holds. If it is false, provide a counterexample. Providing a counterexample here means coming up with a specific example of a set triple A,B,C that violates the statement.

If the bigger set X is clear from the context, one often writes A c for the complement of A in X. Example Recall that the even integers are those integers that are divisible by 2. The odd integers are defined to be those integers that are not even.


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Thus the set of odd integers is the complement of the set of even integers. Proposition Let A, B X. Theorem 5. In words: the complement of the intersection is the union of the complements and the complement of the union is the intersection of the complements. Project Someone tells you that the following equalities are true for all sets A,B,C. In each case, either prove the claim or provide a counterexample. Here are two pictures of De Morgan s equalities. Given sets A 1,A 2,A 3, The set R in Project 5. These logical issues do not cause difficulties in the mathematics discussed in this book.

One would like to go to the other extreme and define a set that contains everything ; however, there is no such set. Is the statement R R true or false? We call a,b an ordered pair. It is the set of all ordered pairs whose first entry is a member of A and whose second entry is a member of B. Example 3, 2 is an ordered pair of integers, and Z Z denotes the set of all ordered pairs of integers. Draw a picture. Proposition Let A,B,C be sets. Decide whether each of the following statements is true or false; in each case prove the statement or give a counterexample.

There is an informal definition and a more abstract definition of the concept of a function. We give both. First Definition. A function consists of a set A called the domain of the function; a set B called the codomain of the function; a rule f that assigns to each a A an element f a B. A useful shorthand for this is f : A B.

Project Discuss how much of this concept coincides with the notion of the graph of f x in your calculus courses. A possible objection to our first definition is that we used the undefined words rule and assigns. To avoid this, we offer the following alternative definition of a function through its graph: This notation suggests that the function f picks up each a A and carries it over to B, placing it precisely on top of an element f a B. Sometimes mathematicians ask whether a function is well defined. What they mean is this: Does the rule you propose really assign to each element of the domain one and only one value in the codomain?

Second Definition. Project Discuss our two definitions of function. What are the advantages and disadvantages of each? Compare them with the definition you learned in calculus. For example, Axiom 1. Review Question. Do you understand the difference between and? Weekly reminder: Reading mathematics is not like reading novels or history.

Matthias Beck Ross Geoghegan. The Art of Proof. Basic Training for Deeper Mathematics

You need to think slowly about every sentence. Usually, you will need to reread the same material later, often more than one rereading. This is a short book. Its core material occupies about pages. Yet it takes a semester for most students to master this material. In summary: read line by line, not page by page. The full. Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction.

Relations are very general thing they are a special type of subset.

The Art of Proof Basic Training for Deeper Mathematics by Matthias Beck and

After introducing. The following points are worth special. Wilson October Chapter 1 Logic Lecture no.