Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross By Herbert I....

Algebra Problems…Solutions

Algebra Problems…Solutions

© 2007 Herbert I. GrossBy Herbert I. Gross and Richard A. Medeiros

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Set 9

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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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The relationship “is less than”

is transitive but neitherreflexive nor symmetric.

Problem #1


Answer: True

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Answer: TrueSolution:

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…

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…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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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 …

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Historical Note

a---- b

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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

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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 < b


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…

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means the same thing as…

a < b b > a

So, for example, to indicate that 3 is less than 4 we could write either

3 < 4 or 4 > 3.

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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)

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(1) a and b are the same point (a = b).

(2) a lies to the left of b (a < b).

(3) a lies to the right of b (a > b).

a

b

a

b

a

b

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…

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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

Key Note


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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.

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next 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”

a < b would mean “a is younger than b”

a ≤ b would mean “a is no older than b”

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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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The relationship “lives next door to”

is transitive.

Problem #2


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Answer: False

Answer: FalseSolution:

To be transitive the following statement must be true…

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That is, A is next to B, and B is next to C, but A is not next to C.

A B C

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“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.

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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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next 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 toB and B lives next door to C then A lives

next door to C.


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A

B C

An example of this is shown below.


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.

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.

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Pictorially…

A B

2

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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.

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The symmetry property tells us that 3 × 5 = 5 × 3.

Problem #3


Answer: False

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To be symmetric the following statement must be true…

“If A = B, then B = A”

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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.

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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


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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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.

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The statement 3 × 5 = 5 × 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.

3

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The relationship “has the same color hair as”

is an equivalence relation.

Problem #4


Answer: True

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Answer: TrueSolution:

In order for the above relation to be an equivalence relation, three things must be true…

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(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 asthe first person. The answer is, “True”.

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Solution:

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Hence, the relation “has the same color hair as” is an equivalence relation.

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(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”.

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Note 4


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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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.

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The relationship “is the sister of”

is symmetric.

Problem #5


Answer: False

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In order for this relation to be symmetric the following statement would have to be true…

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“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.


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.

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Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross By Herbert I....

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