Sets and Logic

Membership
Basics of Set Theory and Logic
S. F. Ellermeyer
August 18, 2000
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.
Examples of Sets
1. A 4,7,13,157.52 is a set with exactly four members. The members of A are the
numbers 4, 7, 13, and 157.52. Hence, we could write
13 A
and
9 A.
2. B 200,190 is the set consisting of all real numbers between 200 and 190 not
including 200 and 190. Hence, we could write
6.34 B
and
214.12 B.
3. C the set of all Presidents of the United States .
4. D the set of all people who were born before 1953
5. E x R|x 2 5x 6 0 is the set of all real numbers which satisfy the equation
x2 5x 6 0.
6. 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
and
0 F.
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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
and
3 G.
Inclusion
If A and 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) and 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 and we write
A B.
If A B and B A, then we say that A equals B and we write A B.
Note that if A B, then there are exactly two possibilities: either A B or A B.
Example
If A 4,7,13,157.52 and B 200,190 , then A B.
Remark: The symbols and are very similar to the symbols and 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 and b are two real numbers and if we
write a b, this means that either a b or a b. Likewise, if A and B are two sets and
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 hand. Thus, a universal set is the “largest possible” set under consideration
within a given discussion. Every set (within the discussion at hand) 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|x2 1 x 3 0 ,
and 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 x2 1 x 3 0. On the
other hand, 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 and we want to define A to
be the set of all real numbers that satisfy x2 1 x 3 0, then we must write
A x R|x2 1 x 3 0 .
This example shows why it is important that the universal set under consideration within a
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particular discussion must be stated or agreed upon beforehand. 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, and x 157.52
and 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 and 5 3i).
Operations
Let A and B be two sets.
The union of A and 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 and B, denoted by A B, is the set of all elements which are both
members of A and 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.
Examples
If A 1,6 and B 4,10,, then A B 1,10 and , A B 4,6 and 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 and 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 and B 12,24 ,
then A B . Any two sets, A and B, for which A B are said to be disjoint.
We remark that the empty set is a subset of every set and that no set is a subset of the
empty set except for the empty set itself.
Below, we list some properties and identities involving the empty set. In each of these, A
stands 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, and B A (for
any sets A and B).
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Figure 1
Below, we list a few properties and identities which hold for any two sets, A and B. These
can be deduced by looking at the Venn diagram in Figure 1.
1. 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 and A A .
9. the universal set
10. A B
A
and the universal set
B and 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 and 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 and 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 and follow almost
immediately by definition.)
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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 and P12 is
false. Abstractly, we can think of P as a function, P : R true, false .
Implications
If P and Q are statements and 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 “” stands for “implies”)
or
P is sufficient for Q
or
Q is necessary for P.
Example
We know that if x 5, then x 2 10, so we can write
x 5 x 2 10.
In other words, if Px is the statement x 5 and 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, 7 2 10 but 7 5. This
example shows that it is possible that the converse of a true implication might not be true.
If P and Q are statements such that P Q and Q P are both true, then we say that
statements P and Q are equivalent and write
P if and only if Q
or
P Q
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or
P is necessary and sufficient for Q.
Example
Let us prove that x 0 and 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 1 2 0
which is true for all real numbers x. By expanding the left hand side of this inequality, we
obtain
x 2 2x 1 0 for all real numbers x
which gives us
x2 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 and its contrapositive are always equivalent.
Example
The contrapositive of the implication
x 5 x 2 10
is the implication
x 2 10 x 5.
More formally, if
then
Px : x 5
Qx : x 2 10
not Px : x 5
not Qx : x 2 10
and we see that the implication P Q is equivalent to the implication not Q not P .
Example
Restate the statement “All real numbers have nonnegative squares” as an implication and
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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 and Logic
We observe the following formal correspondence between logic and set theory: If F and P
are statements that depend on a variable x, then we define the sets
x | Fx is true
and
x | Px is true .
Then, the implication F P is true if and only if .
For example, consider the statements
Fx : x 5
For these statements, we have
Px : x 2 10.
,5
,8
and we observe that F P is true and that . On the other hand, the converse
implication P F is false and, likewise, .
Clearly, two statements F and P are equivalent if and 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 : x2 0
Px : x 6
6
and so .
On the other extreme, if P is a statement that is true for all values of x, then R so, in this
case, the implication F P is true no matter what statement F is. For example, the
implication
is true because for
we have
x 2 9 0 x 2 0
Fx : x2 9 0
Px : x2 0
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3,3
R
and so .
The foregoing discussion shows that an implication F P is false if and 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 and 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 all dogs
C all brown things
Example
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 and weird” is equivalent to M S W and “Some mathematicians are weird” is
equivalent to M W .
Exercises
Consider the pairs of statements, P and Q, given below. For each pair, which of P Q,
Q P, and P Q are true? Recall that we are assuming that the universal set (the
domain of x) isR.
1. Px : x 3
Qx : x 2 9
2. Px : x 3
Qx : x 2 9
3. Px : x 3
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Qx : x2 9
4. Px : x 15
Qx : x 3 18
5. Px, y : x 0 and 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 : There exists M R such that x M for all x A
9. PA,B : A B
QA,B : A
10. Px, y : x y
Qx, y : x z y z for all z R
In the following exercises, let
M all mathematicians
S all smart people
W all weird people
and write the following statements in terms of the sets M, S, and 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 and Q are statements such that P4 is true and 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 and Q are statements such that P4 is true and 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 and Q are statements such that P4 is false and 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 and 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 and 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.
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