Numbers and Their Application to Math and Science

Odd Solutions for HW Numbers Lesson 13

1. Name the axiom used:     10 + 13 + 17 + 23 = 10 + 17 + 13 + 23.
   A. Commutativity for Addition

2. Name the axiom used:     14 • ((17 + 52) + 30) = 14 • (17 + (52 + 30)).
   A.

3. Name the axiom used:     7 × 11 × 13= 11 × 7 × 13.
   A. Commutivity for Multiplication

4. Name the axiom used:     ((7 × 11) × 13) = (7 × (11 × 13)).
   A.

5. Name the axioms used:     x + 0 = x   always.
   A. Additive Identity, Additive Closure

6. Show by counterexample that subtraction is not commutative.
   A.

7. Show by counterexample that subtraction is not associative.
   A. (3 - 1) + 5 = 2 + 5 = 7 3 - (1 + 5) = 3 - 6 = -3

8. Show by counterexample that {negative numbers} are not closed under multiplication.
   A.

9. Show by counterexample that there is no Symmetric Property of Inequality.
   A. -3 < 5 is true, but 5 < -3 is false

10. Show by counterexample that is not transitive.
    A.

11. Is the relationship of "Nicole is a sister of Tom" symmetrical? Show by example why or why not.
    A. No, because Tom is not a sister of Nicole.

    For problems 12-15, which field axioms do the following sets of numbers fail? An example is irrational numbers failing for closure under multiplication since • = 2, which is rational.

12. Natural numbers ().
    A.

13. The integers ().
    A. Inverse for Multiplication (no fractions)

14. The rational numbers ().
    A.

15. The binary digits {0,1} with and as the multiplication type operator and eor as the addition type operator (only difference now is "1+1=0").
    A. None—they are a finite field!! This algebra is known as Boolean Algebra.

16. Consider again the set {0,1} with and and or as operations. Does and distribute over or as well as vice versa? Fill in the table to prove or disprove these distribution rules.
    A.

17. Read sections 3.4 in your geometry textbook. Do problems 3.4: 6, 7, 11, 13, 23.

    6.

    7. a. Solve 2x + 46 < 30.    x < -8.
     b. Properties: Addition and Multiplication Property of Inequality.

    11. Suppose an angle has measure 180 - x and x = 47. Then the angle is 133.
    The Substitution Property is used.

    13. AC = CE thus by Symmetric Property of Equality CE = AC.

    23. Find other relations that satisfy and don't satisfy the transitive property.

    Location directly north satisfies. Is a pupil of does not satisfy.

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