25 Binary Operations
- Submit homework separately from this workbook and staple all pages together. (One staple for the entire submission of all the unit homework)
- Start a new module on the front side of a new page and write the module number on the top center of the page.
- Answers without supporting work will receive no credit.
- Some solutions are given in the solutions manual.
- You may work with classmates but do your own work.
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) |
Exercise 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?