Numbers and Their Application to Math and Science

Homework for Numbers Lesson 13

1. Name the axiom used:     10 + 13 + 17 + 23 = 10 + 17 + 13 + 23.

    

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

    

3. Name the axiom used:     7 × 11 × 13= 11 × 7 × 13.

    

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

   Note: The square root symbol without a vinculum is called a surd. These numbers were originally absurd! Historically this is how radicals were written.

5. Name the axiomS used:     x + 0 = x   always.

    

    

6. Show by counterexample that subtraction is not commutative.

    

7. Show by counterexample that subtraction is not associative.

    

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

    

9. Show by counterexample that there is no Symmetric Property of Greater Than.

    

10. Show by counterexample that is not transitive.

     

11. Is the relationship of "Alexis is a sister of Jason" symmetric? Show by example why or why not.

      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 ().

     

13. The integers ().

     

14. The rational numbers ().

     

15. The binary digits {0,1} with and as the multiplication type operator (×) and eor as the addition type operator (+), the only difference is "1+1=0").

     

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.

    p	q	r	p•(qr)	(p•q)(p•r)	p(q•r)	(pq)•(pr)
    0	0	0
    0	0	1
    0	1	0
    0	1	1
    1	0	0
    1	0	1
    1	1	0
    1	1	1

    
    

17. Read section 3.4 in your geometry textbook. Do problems 3.4: 6 and 22.

     

     

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