1. Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross By Herbert I. Gross and Richard A. Medeiros next Set 9

2. next Tell which of the following statements are true and which are false. In each case explain your choice. © 2007 Herbert I. Gross

3. next The relationship “is less than” is transitive but neither reflexive nor symmetric. Problem #1 © 2007 Herbert I. Gross Answer: True next

4. 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. next © 2007 Herbert I. Gross

5. Solution: If the relation were symmetric… next © 2007 Herbert I. Gross …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. next

6. 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. next © 2007 Herbert I. Gross

7. 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 … next Historical Note a -- b next

8. © 2007 Herbert I. Gross 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… next ab 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

9. © 2007 Herbert I. Gross 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… next means the same thing as… a < bb > a So, for example, to indicate that 3 is less than 4 we could write either 3 < 4 or 4 > 3. next

10. 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… © 2007 Herbert I. Gross (1) a is equal to b (a = b) (2) a is less than b (a < b) (3) a is greater than b (a > b) next

11. © 2007 Herbert I. Gross (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… next

12. 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

13. © 2007 Herbert I. Gross 1 next 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.

14. 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… © 2007 Herbert I. Gross 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” next a > b would mean “a is older than b” a ≥ b would mean “a is at least as old as b”

15. The relationship “lives next door to” is transitive. Problem #2 © 2007 Herbert I. Gross next Answer: False

16. Solution: To be transitive the following statement must be true… next © 2007 Herbert I. Gross That is, A is next to B, and B is next to C, but A is not next to C. ABC next “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.

17. Note 2 Notice the use of the word “must” in the statement… © 2007 Herbert I. Gross “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. next

18. 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. © 2007 Herbert I. Gross next ABC An example of this is shown below.

19. © 2007 Herbert I. Gross 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. next

20. © 2007 Herbert I. Gross Pictorially… AB 2 next 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.

21. The symmetry property tells us that 3 × 5 = 5 × 3. Problem #3 © 2007 Herbert I. Gross Answer: False next

22. Answer: False Solution: To be symmetric the following statement must be true… “If A = B, then B = A” next © 2007 Herbert I. Gross 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. next The point is that the symmetric property of equality does not establish the truth of the statement A = B. next

23. Note 3 © 2007 Herbert I. Gross 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. next 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.

24. © 2007 Herbert I. Gross 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

25. next The relationship “has the same color hair as” is an equivalence relation. Problem #4 © 2007 Herbert I. Gross Answer: True next

26. Answer: True Solution: In order for the above relation to be an equivalence relation, three things must be true… next © 2007 Herbert I. Gross (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”. next

27. Solution: next © 2007 Herbert I. Gross Hence, the relation “has the same color hair as” is an equivalence relation. next (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”.

28. Note 4 © 2007 Herbert I. Gross 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. next 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.

29. The relationship “is the sister of” is symmetric. Problem #5 © 2007 Herbert I. Gross Answer: False next

30. Answer: False Solution: In order for this relation to be symmetric the following statement would have to be true… next © 2007 Herbert I. Gross “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. next

31. Note 5 © 2007 Herbert I. Gross 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.

32. next © 2007 Herbert I. Gross 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.

33. 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. next