32 Homework

Question #1

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. [latex]5 * 9[/latex]                    b. [latex]4 * 8[/latex]                    c. [latex]b * a[/latex]                    d. [latex]1 ◊ 5[/latex]

e. [latex]7 ◊ 4[/latex]                      f. [latex]b ◊ a[/latex]                      g. [latex]88 ∇ 79[/latex]               h. [latex]m ∇ n[/latex]

i. [latex]b ∇ a[/latex]                       j. [latex]6 λ 5[/latex]                      k. [latex]2 λ 7[/latex]                      l. [latex]b λ a[/latex]

m. [latex]8\dagger1[/latex]                   n. [latex]4\dagger6[/latex]                    o. [latex]b\dagger a[/latex]                     p. [latex]8 @ 5[/latex]

q. [latex]9 @ 4[/latex]                     r. [latex]b @ a[/latex]                      s. [latex]6 Θ 7[/latex]                      t. [latex]2 Θ 5[/latex]

u. [latex]b  Θ  a[/latex]                     v. [latex]3\otimes8[/latex]                    w. [latex]5 \otimes 6[/latex]                  x. [latex]b \otimes a[/latex]

y. [latex]7 ⊥ 8[/latex]                  z. [latex]3 ⊥ 2[/latex]                    aa. [latex]b ⊥ a[/latex]

Question #2

Simplify each of the following. Show each step.

a. [latex]3 * (4 * 2)[/latex]                    b. [latex]3 @ (5 @ 2)[/latex]                    c. [latex]3 ∇ (5 ∇ 2)[/latex]

Question #3

Compute the following, using the definitions for the operations defined in the "Operations and Properties" section of this book. Show each step.

a. [latex](1 \otimes 2) Θ 7[/latex]                    b. [latex](3 * 1) ◊ c[/latex]                    c. [latex](8 ∇ 5) λ 6[/latex]

d. [latex]7 \otimes (6 ◊ 13)[/latex]                  e. [latex]5 * (2 Θ 2)[/latex]                    f. [latex]8 ∇ (5 λ 6)[/latex]

Question #4

Suppose that a number system uses only the symbols a, b, c, d, and the operation [latex]Ψ[/latex].  The basic facts are illustrated in the table.

Ex.    A [latex]Ψ[/latex] C = B     or    D [latex]Ψ[/latex] B = B

b)  Is the operation commutative? Why?

c)  Does the operation have an identity? If so, what is it?