11 KiB
Example 1.1.1
Page 24
Use variables to rewrite the following sentences more formally.
a. Are there numbers with the property that the sum of their squares equals the square of their sum?
b. Given any real number, its square is nonnegative.
Solution
a. Are there numbers a and b with the property that a^2 + b^2 = (a + b)^2?
Or: Are there numbers $a$$ and b such that a^2 + b^2 = (a + b)^2?
Or: Do there exist any numbers a and b such that a^2 + b^2 = (a + b)^2?
b. Given any real number r, r^2 is nonnegative.
Or: For any real number r, r^2 \geq 0.
Or: For every real number r, r^2 \geq 0.
Example 1.1.2
Page 26
Fill in the blanks to rewrite the following statement:
For every real number x, if x is nonzero then x^2 is positive.
a. If a real number is nonzero, then its square ________.
b. For every nonzero real number x, ________.
c. If x ________, then ________.
d. The square of any nonzero real number is ________.
e. All nonzero real numbers have ________.
Solution.
a. is positive.
b. x^2 is positive.
c. is a nonzero real number, x^2 is positive.
d. positive.
e. positive squares .
Example 1.1.3
Page 27
Fill in the blanks to rewrite the following statement: Every pot has a lid.
a. All pots ________.
b. For every pot P, there is ________.
c. For every pot P, there is a lid L such that ________.
Solution
a. have lids.
b. a lid.
c. L is a lid for P..
Example 1.1.4
Page 28
Fill in the blanks to rewrite the following statement in three different ways:
There is a person in my class who is at least as old as every person in my clas.
a. Some ________ is at least as old as ________.
b. There is a person p in my class such that p is ________.
c. There is a person p in my class with the property that for every person q
in my class, p is ________.
Solution
a. person; every person.
b. at least as old as every person in my class.
c. at least as old as q.
Example 1.2.1
Page 30
Using the Set-Roster Notation
a. Let A = \{1, 2, 3\}, B = \{3, 1, 2\}, and C = \{1, 1, 2, 3, 3, 3\}.
What are the elements of A, B, and C? How are A, B, and C related?
b. Is \{0\} = 0?
c. How many elements are in the set \{1, \{1}\}?
d. For each nonnegative integer n, let U_n = \{n, -n\}. Find U_1, U_2,
and U_0.
Solution
a. Let A = \{1, 2, 3\}, B = \{3, 1, 2\}, and C = \{1, 1, 2, 3, 3, 3\}.
What are the elements of A, B, and C? How are A, B, and C related?
A, B, and C have exactly the same three elements, 1, 2, and 3.
Therefore, A, B, and C are simply different ways to represent the same
set.
b. Is \{0\} = 0?
\{0\} \neq 0 because \{0\} is a set with one element, namely 0, whereas
0 is just the symbol that represents the number zero.
c. How many elements are in the set \{1, \{1}\}?
The set \{1, \{1\}\} has two elements. 1 and the set whose only element is
1.
d. For each nonnegative integer n, let U_n = \{n, -n\}. Find U_1, U_2,
and U_0.
U_1 = \{1, -1\}, \quad U_2 = \{2, -2\}, \quad U_0 = \{0, 0\} = \{0\}
Example 1.2.2
Page 31
Using the Set-Builder Notation
Given that \mathbb{R} denotes the set of all real numbers, \mathbb{Z} the
set of all integers, and \mathbb{Z}^+ the set of all positive integers,
describe each of the following sets.
a. \{x \in \mathbb{R} | -2 < x < 5\}
b. \{x \in \mathbb{Z} | -2 < x < 5\}
c. \{x \in \mathbb{Z}^+ | -2 < x < 5\}
Solution
a. \{x \in \mathbb{R} | -2 < x < 5\}
\{x \in \mathbb{R} | -2 < x < 5\} is the open interval of real numbers
(strictly) between -2 and 5. It is pictured as follows (see page 31).
b. \{x \in \mathbb{Z} | -2 < x < 5\}
\{x \in \mathbb{Z} | -2 < x < 5\} is the set of all integers (strictly)
between -2 and 5. It is equal to the set \{-1, 0, 1, 2, 3, 4}.
c. \{x \in \mathbb{Z}^+ | -2 < x < 5\}
Since all the integers in \mathbb{Z}^+ are positive,
\{x \in \mathbb{Z}^+ | -2 < x < 5\} = \{1, 2, 3, 4\}.
Example 1.2.3
Page 32
Let A = \mathbb{Z}^+, B = \{n \in \mathbb{Z} | 0 \leq n \leq 100\}, and
C = \{100, 200, 300, 400, 500\}. Evaluate the truth and falsity of each of the
following statements
a. B \subseteq A
b. C is a proper subset of A.
c. C and B have at least one element in common
d. C \subseteq B
e. C \subseteq C
Solution
a. B \subseteq A
False. Zero is not a positive integer. Thus zero is in B but zero is not in
A, and so B \nsubseteq A
b. C is a proper subset of A.
True. Each element in C is a positive integer, and hence, is in A, but there
are elements in A that are not in C. For instance, 1 is in A and not in
C.
c. C and B have at least one element in common
True. For example, 100 is in both C and B.
d. C \subseteq B
False. For example, 200 is in C but not in B.
e. C \subseteq C
True. Every element in C is in C. In general, the definition of a subset
implies that all sets are subsets of themselves.
Example 1.2.4
Page 33
Distinction between \in and $\subseteq$
Which of the following are true statements?
a. 2 \in \{1, 2, 3\}
b. \{2\} \in \{1, 2, 3\}
c. 2 \subseteq \{1, 2, 3\}
d. \{2\} \subseteq \{1, 2, 3\}
e. \{2\} \subseteq \{\{1\}, \{2\}\}
f. \{2} \in \{\{1\}, \{2\}\}
Solution
Only (a), (d), and (f) are true.
For (b) to be true, the set \{1, 2, 3\} would have to contain the element
\{2\}. But the only elements of \{1, 2, 3\} are 1, 2, and 3, and 2
is not equal to \{2\}. Hence (b) is false.
For c to be true, the number 2 would have to be a set and every element in
the set 2 would have to be an element of \{1, 2, 3}. This is not the case,
so c is false.
For (e) to be true, every element in the set containing only the number 2
would have to be an element of the set whose elements are \{1\} and \{2\}.
But 2 is not equal to either \{1\} or \{2\}, and so (e) is false.
Example 1.2.5 Ordered Pairs
Page 34
a. Is (1, 2) = (2, 1)?
b. Is \left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)?
c. What is the first element of (1, 1)?
Solution
a. Is (1, 2) = (2, 1)?
No, By definition of equality of ordered pairs,
(1, 2) = (2, 1) if, and only if, 1 = 2, and 2 = 1.
But 1 \neq 2, and so the ordered pairs are not equal.
b. Is \left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)?
Yes. By definition of equality of ordered pairs,
\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right) if, and
only if, 3 = \sqrt{9} and \dfrac{5}{10} = \dfrac{1}{2}.
Because these equations are both true, the ordered pairs are equal.
c. What is the first element of (1, 1)?
In the ordered pair (1, 1), the first and second elements are both 1.
Example 1.2.6 Ordered $n$-tuples
Page 34
a. Is (1, 2, 3, 4) = (1, 2, 4, 3)?
b. Is
\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)?
Solution
a. Is (1, 2, 3, 4) = (1, 2, 4, 3)?
No. By definition of equality of ordered 4-tuples,
(1, 2, 3, 4) = (1, 2, 4, 3) \leftrightarrow 1 = 1, 2 = 2, 3 = 4, and 4 = 3 But 3 \neq 4, and so the ordered 4-tuples are not equal.
b. Is
\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)?
Yes. By definition of equality of ordered triples.
\left(3, (-2)^2, \frac{1}{2}\right) = \left(\sqrt{9}, 4, \frac{3}{6}\right) \leftrightarrow 3 = \sqrt{9} \text{ and } (-2)^2 = 4 \text{ and } \frac{1}{2} = \frac{3}{6} Because these equations are all true, the two ordered triples are equal.
Example 1.2.7 Cartesian Products
Page 35
Let A = \{x, y\}, B = \{1, 2, 3\}, and C = \{a, b\}.
a. Find A \times B.
b. Find B \times A.
c. Find A \times A.
d. How many elements are in A \times B, B \times A, and A \times A?
e. Find (A \times B) \times C
f. Find A \times B \times C
g. Let \mathbb{R} denote the set of all real numbers. Describe
\mathbb{R} times \mathbb{R}.
Solution
a. Find A \times B.
A \times B = \{(x, 1), (y, 1), (x, 2), (y, 2), (x, 3), (y, 3)\} b. Find B \times A.
B \times A = \{(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)\} c. Find A \times A.
A \times A = \{(x, x), (x, y), (y, x), (y, y)\} d. How many elements are in A \times B, B \times A, and A \times A?
A \times B has 6 elements. Note that this is the number of elements in A
times the number of elements in B. B \times A has 6 elements, the number of
elements in B times the number of elements in A. A \times A has 4
elements, the number of elements in A times the number of elements in A.
e. Find (A \times B) \times C
(A \times B) \times C = \{(u, v) | u \in A \times B \text{ and } v \in C\} By definition of Cartesian product.
(A \times B) \times C = \{((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a), ((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b), ((y, 2), b), ((y, 3), b)\} f. Find A \times B \times C
The Cartesian product A \times B \times C is superficially similar to but is
not quite the same mathematical object as (A \times B) \times C.
(A \times B) \times C is a set of ordered pairs of which one element is itself
an ordered pair, whereas A \times B \times C is a set of ordered triples. By
definition of Cartesian product,
A \times B \times C = \{(u, v, w) | u \in A, v \in B, \text{ and } w \in C\} A \times B \times C = \{(x, 1, a), (x, 2, a), (x, 3, a), (y, 1, a), (y, 2, a), (y, 3, a), (x, 1, b), (x, 2, b), (x, 3, b), (y, 1, b), (y, 2, b), (y, 3, b)\} g. Let \mathbb{R} denote the set of all real numbers. Describe
\mathbb{R} times \mathbb{R}.
\mathbb{R} \times \mathbb{R} is the set of all ordered pairs (x, y) where
both x and y are real numbers. If horizontal and vertical axes are drawn on
a plane and a unit length is marked off, then each ordered pair in
\mathbb{R} \times \mathbb{R} corresponds to a unique point in the plane, with
the first and second elements o the pair indicating, respectively, the
horizontal and vertical positions of the point. The term Cartesian plane is
often used to refer to a plane with this coordinate system, as illustrated in
Figure 1.2.1 (see page 36).
Example 1.2.8 Strings
Page 36
Let A = \{a, b\}. List all the strings of length 3 over A with at least two
characters that are the same.
Solution
aab, aba, baa, aaa, bba, bab, abb, bbb
In computer programming it is important to distinguish among different kinds of data structures and to respect the notations that are used for them. Similarly in mathematics, it is important to distinguish among, say, {a, b, c}, {{ab}, c}, (a, b, c), (a, (b, c)), abc and so forth, because these are all significantly different objects.