Basics of Set Theory and Logic. Set Theory

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1 Basics of Set Theory Logic S. F. Ellermeyer August 18, 2000 Membership Set Theory A set is a well-defined collection of objects. Any object which is in a set is called a member of the set. If the object x is a member of the set A, then we write x A which is read as x is a member of A or xbelongs toa or xis in A or xis an element of A. If the object x is not a member of A, then we write x A. s of Sets 1. A 4,7,13, is a set with exactly four members. The members of A are the numbers 4, 7, 13, Hence, we could write 13 A 9 A. 2. B 200,190 is the set consisting of all real numbers between not including Hence, we could write 6.34 B B. 3. C the set of all Presidents of the United States. 4. D the set of all people who were born before E x R x 2 5x 6 0 is the set of all real numbers which satisfy the equation x 2 5x F x R x 14 4 x is the set of all real numbers which satisfy the inequality x 14 4 x. Hence, we could write 10 F 0 F. 1

2 7. G n N n is a divisor of 275 is the set of all natural numbers which are divisors of the number 275. Note that 11 G 3 G. Inclusion If A B are two sets such that every member of A is also a member of B, then we say that A is a subset of B (or that A is included in B) we write A B. If A B but B has members which are not members of A, then we say that A is a proper subset of B we write A B. If A B B A, then we say that A equals B we write A B. Note that if A B, then there are exactly two possibilities: either A B or A B. If A 4, 7, 13, B 200, 190, then A B. Remark: The symbols are very similar to the symbols which are used in describing the order relation of two real numbers. This can help us remember what the set inclusion symbols mean. In particular, if a b are two real numbers if we write a b, this means that either a b or a b. Likewise, if A B are two sets we write A B, this means that either A B or A B. Universal Sets A universal set is the set that is assumed to contain all members pertaining to the discussion at h. Thus, a universal set is the largest possible set under consideration within a given discussion. Every set (within the discussion at h) is assumed to be a subset of the universal set. In this course, the universal set will almost always be assumed to be the set of real numbers, R. Hence, if we define the set A x x 2 1 x 3 0, if it is understood that the universal set is R, then we can conclude that A 3 because 3 is the only real number that satisfies the equation x 2 1 x 3 0. On the other h, if we are assuming that the universal set is C (the set of all complex numbers), then we have A 3, i, i. If the universal is understood to be C we want to define A to be the set of all real numbers that satisfy x 2 1 x 3 0, then we must write A x R x 2 1 x 3 0. This example shows why it is important that the universal set under consideration within a 2

3 particular discussion must be stated or agreed upon beforeh. Unless specifically stated otherwise, we will always assume that the universal set is R. The Complement of a Set If A is a set, then the complement of A, denoted by A, is the set of all non members of A. For example if A 4,7,13,157.52, then A x x 4, x 7, x 13, x if A 4,, then A,4. Note the importance of knowing what the universal set is in determining A. For example, if we were using the set of complex numbers as the universal set, then the complement of A 4, would include the interval, 4 as well as all of the imaginary numbers (such as i 5 3i). Operations Let A B be two sets. The union of A B, denoted by A B, is the set of all elements which are either members of A or members of B. The intersection of A B, denoted by A B, is the set of all elements which are both members of A members of B. The difference, A B (sometimes written as A/B), is the set of all elements which are members of A but not members of B. s If A 1,6 B 4,10,, then A B 1,10, A B 4,6 A B 1,4. The Empty Set The empty set is defined to be the set which has no members. The empty set is denoted by the symbol or by. Other names for the empty set are the null set the vacuous set. The concept of the empty set is necessary because we need to be able to describe situations where two given sets do not intersect. For example, if A 1, 6 B 12, 24, then A B. Any two sets, A B, for which A B are said to be disjoint. We remark that the empty set is a subset of every set that no set is a subset of the empty set except for the empty set itself. Below, we list some properties identities involving the empty set. In each of these, A sts for any given set. 1. A 2. A A 3. A 4. A A 5. A The Venn diagram in Figure 1 shows a general picture relating A B, A B, B A (for any sets A B). 3

4 Figure 1 Below, we list a few properties identities which hold for any two sets, A B. These can be deduced by looking at the Venn diagram in Figure A A B A B 2. A B A B A B B A 3. A A B 4. A B A 5. A B A 6. A B A B 7. A B B 8. A A the universal set A A. 9. the universal set the universal set 10. A B A B A B A B (These are called DeMorgan s Laws.) Sample Proof Let us prove the equality A A B A B. To prove this equality, we must prove that A A B A B that A B A B A. First, we prove that A A B A B. Let x A. Clearly, either x B or x B.Ifx B, then x A B if x B, then x A B. We conclude that either x A B or x A B. Thus x A B A B. This shows that A A B A B. Next, we prove that A B A B A. Let x A B A B. Then either x A B or x A B. In either case, it must be true that x A. This shows that A B A B A. Exercises Prove statements 2 through 10 above. (Some of the proofs are very short follow almost immediately by definition.) 4

5 Logic Mathematical Statements A mathematical statement that depends on a variable, x, is a statement pertaining to x which is either true or false, depending on the value of x. For example, if we write Px : x 5, then we are saying that Px is the statement x is less than 5. Note that, depending on the value of x, this statement is either true or false. For example, P4.6 is true P12 is false. Abstractly, we can think of P as a function, P : R true, false. Implications If P Q are statements if we want to assert that Qx must be true whenever Px is true, then we say: If P, then Q or P Q (where the symbol sts for implies ) or P is sufficient for Q or Q is necessary for P. We know that if x 5, then x 2 10, so we can write x 5 x In other words, if Px is the statement x 5 Qx is the statement x 2 10, then P Q is true because Qx is true whenever Px is true. The Converse of an Implication The converse of an implication P Q is the implication Q P. For example, the converse of the implication in the preceding example is x 2 10 x 5. Note that this implication is not true because, for instance, but 7 5. This example shows that it is possible that the converse of a true implication might not be true. If P Q are statements such that P Q Q P are both true, then we say that statements P Q are equivalent write or P if only if Q P Q 5

6 or P is necessary sufficient for Q. Let us prove that x 0 x 1/x 2 are equivalent: First, observe that if x 0, then 1/x 0 so x 1/x 0. Also, if x 0, then x 1/x is not even defined. We conclude that if x 0, then it certainly is not true that x 1/x 2. In other words, if x 1/x 2, then it must be true that x 0. This establishes the truth of the implication x 1 x 2 x 0. To prove the converse, we consider the inequality x which is true for all real numbers x. By exping the left h side of this inequality, we obtain x 2 2x 1 0 for all real numbers x which gives us x 2 1 2x for all real numbers x. If x 0, then we can divide both sides of the preceding inequality by x (without reversing the order of the inequality) to obtain x 1 x 2. This establishes the truth of the implication x 0 x 1 x 2. The Contrapositive of an Implication The contrapositive of an implication P Q is the implication not Q not P.An implication its contrapositive are always equivalent. The contrapositive of the implication x 5 x 2 10 is the implication More formally, if then x 2 10 x 5. Px : x 5 Qx : x 2 10 not Px : x 5 not Qx : x 2 10 we see that the implication P Q is equivalent to the implication not Q not P. Restate the statement All real numbers have nonnegative squares as an implication 6

7 state the contrapositive of this implication. Original Statement: If x is a real number, then x 2 0. Contrapositive: If x 2 0, then x is not a real number. Set Theory Logic We observe the following formal correspondence between logic set theory: If F P are statements that depend on a variable x, then we define the sets x Fx is true x Px is true. Then, the implication F P is true if only if. For example, consider the statements Fx : x 5 Px : x For these statements, we have,5,8 we observe that F P is true that. On the other h, the converse implication P F is false, likewise,. Clearly, two statements F P are equivalent if only if. If F is a statement that is not true for any value of x, then so, in this case, the implication F P is true no matter what statement P is! For example, the implication is true because for we have x 2 0 x 6 Fx : x 2 0 Px : x 6 6 so. On the other extreme, if P is a statement that is true for all values of x, then Rso, in this case, the implication F P is true no matter what statement F is. For example, the implication x x 2 0 is true because for Fx : x Px : x 2 0 we have 7

8 3,3 R so. The foregoing discussion shows that an implication F P is false if only if there exists a real number, x, such that Fx is true but Px is false. For example, the implication x 2 10 x 5 is false because for Fx : x 2 10 Px : x 5 we have F7 true but P7 false. Further Connections Between Sets Logic 1. Statements containing for all can be stated as set inclusions. For example, the statement All cows are white can be stated as A B where A all cows B all white things 2. Statements containing there exists can be stated in terms of the empty set. For example, the statement Some dogs are brown can be stated as A where A all brown dogs or as B C where B C all dogs all brown things Consider the statement All mathematicians are either smart or weird. If we let M all mathematicians S all smart people W all wierd people, then the above statement is equivalent to M S W. Likewise, All mathematicians are smart weird is equivalent to M S W Some mathematicians are weird is equivalent to M W. Exercises Consider the pairs of statements, P Q, given below. For each pair, which of P Q, Q P, P Q are true? Recall that we are assuming that the universal set (the domain of x) isr. 1. Px : x 3 Qx : x Px : x 3 Qx : x Px : x 3 8

9 Qx : x Px : x 15 Qx : x Px, y : x 0 y 0 x Qx, y : y y x 2 6. Px : x for all 0 Qx : x 0 7. Px : x 4 Qx : 4 x 4 8. PA : x 46 for all x A QA : 9. PA,B : A B QA,B : A 10. Px, y : x y There exists M R such that x M for all x A Qx, y : x z y z for all z R In the following exercises, let all mathematicians M S W all smart people all weird people write the following statements in terms of the sets M, S, W. 1. All mathematicians are smart. 2. Some mathematicians are smart. 3. Some people who are smart or weird are mathematicians. 4. All smart people who are not weird are not mathematicians. 5. Some weird mathematicians are smart. Answer the following questions: 1. Suppose that P Q are statements such that P4 is true Q4 is false. Is the implication P Q true or false or can t this be determined? What about the implication Q P? Give examples. 2. Suppose that P Q are statements such that P4 is true Q4 is true. Is the implication P Q true or false or can t this be determined? What about the implication Q P? Give examples. 3. Suppose that P Q are statements such that P4 is false Q4 is false. Is the implication P Q true or false or can t this be determined? What about the implication Q P? Give examples. 4. Suppose that P Q are statements such that Qx is true for all x. Is the implication P Q true or false or can t this be determined? What about the implication Q P? Give examples. 5. Suppose that P Q are statements such that Qx is false for all x. Is the implication P Q true or false or can t this be determined? What about the implication Q P? Give examples. 9