Download Lesson 9 Solutions - Adjective Noun Math

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Algebra
Problems…
Solutions
Set 9
© 2007 Herbert I. Gross
By Herbert I. Gross and Richard A. Medeiros
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Tell which of the following
statements are true and which
are false. In each case explain
your choice.
© 2007 Herbert I. Gross
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Problem #1
The relationship
“is less than”
is transitive but neither
reflexive nor symmetric.
Answer: True
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Answer: True
Solution:
If the relation “is less than” were reflexive
it would mean that for any number n, n
would be less than n. Since no number
can be less than itself, the relation is not
reflexive.
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Solution:
If the relation were symmetric…
…then if the first number was less than
the second number, then the second
number would also have to be less than
the first number. However, if the first
number is less than the second number,
that means that the second number is
greater than the first number. Hence, the
relation is not symmetric.
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Solution:
On the other hand: if the first number is
less than the second number and the
second number is less than the third
number, then the first number is also less
than the third number.
Hence, the relation is transitive.
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Historical Note
The symbolism behind the equal sign is
that it consists of two parallel lines,
signifying that the distance between them
is the same. Hence, if the numbers a and b
are at opposite ends of the equal sign, the
equal spacing between the two lines
symbolized that the two numbers were
equal. That is …
-a -© 2007 Herbert I. Gross
b
Then to indicate that a is less than b, the
two lines of the equal sign were “pinched”
together beside the a to indicate that the
lesser number was next to the smaller space.
That is…
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a
b
Eventually the two lines were closed at the
smaller space to eliminate any possible
ambiguity; thus the lines form an arrowhead,
with the point of the arrow pointing to the
lesser number. That is…
a
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<
b
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We may then read the diagram (a < b)
either as “a is less than b” or as “b is
greater than a” (in either case, the arrow
“points” to the smaller number). Thus…
a
<
b
means the same
thing as…
b
>
So, for example, to indicate that 3 is
less than 4 we could write either
3 < 4 or 4 > 3.
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a
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Note 1
As yet we have not talked about the rule
of trichotomy which involves equalities
and inequalities. The rule states that for
any two numbers a and b; exactly one of
the following three statements is true…
(1) a is equal to b (a = b)
(2) a is less than b (a < b)
(3) a is greater than b (a > b)
© 2007 Herbert I. Gross
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1 In terms of the number line
wherein we treat a and b as points, the rule
of trichotomy says that exactly one of the
following three statements is true…
(1) a and b are the same
point
(a
=
b).
b
a
(2) a lies to the left of b (a < b).
b
a
(3) a lies to the right of b (a > b).
b
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a
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Key Note
Therefore, if it's true that
a<b
(or equivalently b > a),
then it is false that
a>b
(or, equivalently, b < a).
© 2007 Herbert I. Gross
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1 There is a subtle difference between
saying, for example, “x is less than 3” and
“x is no greater than 3”. The statement “x is
no greater than 3” means that x is either less
than 3 or it's equal to 3. The symbol for
expressing “less than or equal to” is ≤.
Thus while it's false that 3 < 3, it is true that
3 ≤ 3 (that is, it's true that 3 is no greater
than 3). Thus, the relation “is no greater
than” is reflexive but not symmetric. That
is: if it's true that a is no greater than b, it's
also true that b is not less than a.
© 2007 Herbert I. Gross
If you are still not comfortable working
with the equality and inequality of numbers,
work instead with a relation such as…
“is the same age as”. Notice that “a is the
same age as b” is an equivalence relation;
and in this context…
a = b would mean “a is the same age as b”
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a < b would mean “a is younger than b”
a ≤ b would mean “a is no older than b”
a > b would mean “a is older than b”
a ≥ b would mean “a is at least as old as b”
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Problem #2
The relationship
“lives next door to”
is transitive.
Answer: False
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Answer: False
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Solution:
To be transitive the following statement
must be true…
“If A lives next door to B and B lives next
door to C, then A must live next door to C”.
However, as shown in the diagram below
this need not be true.
A
B
C
That is, A is next to B, and B is next to C,
but A is not next to C.
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Note 2
Notice the use of the word “must” in the
statement…
“If A lives next door to B, and B lives next
door to C, then A must live next door to C”.
The “must” says that once we know
that A lives next door to B, and that
B lives next door to C; it follows
inescapably that A lives next door to C.
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So, for example, it is possible for three
houses to be arranged in such a way that it
is true that if A lives next door to
B and B lives next door to C then A lives
next door to C.
An example of this is shown below.
A
B
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C
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2 However, for a relation to be transitive,
it's not enough that there are times when the
conditions are met. Rather there has to be
no way for the conditions not to be met.
In the above sense, we also see that the
relationship is not reflexive. That is (even
though there is a possibility that the person
owns two adjacent houses), a person doesn't
necessarily live next door to himself.
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2 On the other hand, the relation is
symmetric because if A lives next door to B,
it has to follow that B also lives next door to
A. That is, no matter how we visualize the
statement that A lives next door to B, it
follows inescapably that B also lives next
door to A.
Pictorially…
A
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B
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Problem #3
The symmetry property tells us
that 3 × 5 = 5 × 3.
Answer: False
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Answer: False
Solution:
To be symmetric the following statement
must be true…
“If A = B, then B = A”
Therefore, if we let A represent 3 × 5, and we
let B represent 5 × 3, the symmetric property
would say that if 3 × 5 = 5 × 3, then it would
also be true that 5 × 3 = 3 × 5.
The point is that the symmetric property of
equality does not establish
the truth of the statement A = B.
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Note 3
It's important to understand the meaning of
“if”. For example, 8 is not equal to 3.
However, in terms of the symmetric
property, if it had been true that 8 = 3, then
it would also have been true that 3 = 8.
As a non-mathematical example, the
statement “If it rains, I'll go to the movies”
says nothing about whether it will or will
not rain; but only what will happen if it
does rain.
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The statement 3 × 5 = 5 × 3
3
concerns a property of
multiplication. More formally, the
truth of this statement is called the
commutative property of multiplication.
The next lesson will deal with the
properties of addition and multiplication,
but in this lesson we are focusing on the
properties of equality.
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Problem #4
The relationship
“has the same color hair as”
is an equivalence relation.
Answer: True
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Answer: True
Solution:
In order for the above relation to be an
equivalence relation, three things must be
true…
(1) Each person must have the same
color hair as him or herself.
The answer is, “True”.
(2) If the first person has the same color
hair as the second person, the second
person must have the same color hair as
the first person. The answer is, “True”.
© 2007 Herbert I. Gross
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Solution:
(3) If the first person has the same color
hair as the second person and if the
second person has the same color hair
as the third person, the first person
must have the same color hair as the
third person.
The answer is “True”.
Hence, the relation “has the same color
hair as” is an equivalence relation.
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Note 4
Don't read more into an equivalence relation
than what is there. For example, all we've
shown is that if two people have the same color
hair, we cannot distinguish between them with
respect to the color of their hair. It doesn't
imply that they share any other characteristics.
Another example: when the Declaration of
Independence refers to all men being created
equal, it doesn't mean with respect to wealth
or appearance but rather, that they are equal in
the eyes of the law.
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Problem #5
The relationship
“is the sister of”
is symmetric.
Answer: False
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Answer: False
Solution:
In order for this relation to be symmetric the
following statement would have to be true…
“If A is the sister of B, then B is the sister
of A.”
To show that this statement is not always
true, just suppose A is a girl and that B is a
boy. Thus, if it's true that Mary is the sister
of William, William is the brother (not the
sister) of Mary.
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Note 5
Sometimes we restrict the set of objects
to which the relationship applies.
For example, suppose we were only
considering women with respect to the
relation “is the sister of”. In this case, it
would be true that if A is the sister of B,
then B is also the sister of A. In other
words, the relation is symmetric, since we
restricted our attention to women only.
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5
As we saw with the relationship
“is less than”, the fact that a relationship
isn't symmetric doesn't mean that it can't
be transitive. In the same way, “is the
sister of” is not symmetric, but it is
transitive, even if C is a male. That is,
suppose A represents Mary, B represents
Jane and C represents William. It is true
that if Mary is the sister of Jane and Jane is
the sister of William, then Mary is also the
sister of William.
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5
Sometimes there are exceptional
circumstances that we might not have
thought about. For example, if Mary and
Jane have different fathers, it's possible
that she is William's sister, but Mary isn't.
In such an extreme case, we might want to
emend the transitive property to say “If A,
B, and C are from the same household...”.
However, such subtleties will not occur in
this course where we'll be primarily
concerned with equalities and inequalities.
© 2007 Herbert I. Gross