25 Homework
Compute and simplify the following. Show all of the steps. Use the binary operations defined in the “Operations and Properties” section of this book.
| a. 5 * 9 | b. 4 * 8 | c. b * a | d. 1 ◊ 5 |
| e. 7 ◊ 4 | f. b ◊ a | g. 88 ∇ 79 | h. m ∇ n |
| i. b ∇ a | j. 6 λ 5 | k. 2 λ 7 | l. b λ a |
| m. 8 [latex]\dagger[/latex] 1 | n. 4 [latex]\dagger[/latex] 6 | o. b [latex]\dagger[/latex] a | p. 8 @ 5 |
| q. 9 @ 4 | r. b @ a | s. 6 Θ 7 | t. 2 Θ 5 |
| u. b Θ a | v. 3 [latex]\otimes[/latex] 8 | w. 5 [latex]\otimes[/latex] 6 | x. b [latex]\otimes[/latex] a |
| y. 7 ⊥ 8 | z. 3 ⊥ 2 | aa. b ⊥ a |
Simplify each of the following. Show each step.
| a. 3 * (4 * 2) | b. 3 @ (5 @ 2) | c. 3 ∇ (5 ∇ 2) |
Compute the following, using the definitions for the operations defined in the “Operations and Properties” section of this book. Show each step.
| a. (1 [latex]\otimes[/latex] 2) Θ 7 | b. (3 * 1) ◊ c | c. (8 ∇ 5) λ 6 |
| d. 7 [latex]\otimes[/latex] (6 ◊ 13) | e. 5 * (2 Θ 2) | f. 8 ∇ (5 λ 6) |
4
Suppose that a number system uses only the symbols a, b, c, d, and the operation Ψ. The basic facts are illustrated in the table.
Ex. A Ψ C = B or D Ψ B = B
| Ψ | A | B | C | D |
| A | C | D | B | A |
| B | D | A | C | B |
| C | B | C | B | C |
| D | A | B | C | D |
a) Is the system closed? Why?
b) Is the operation commutative? Why?
c) Does the operation have an identity? If so, what is it?