Sup-pose that three books numbered 1, 2, and 3 are placed next to one another on a shelf. If we remove volume 3 and place it before volume 1, the new order of books is 3, 1, 2. Let’s call this replacement R. We can write

which indicates the books were switched in order from 1, 2, 3 to 3, 1, 2. Other possible replacements are S, T, U, V, and I, as indicated.

Replacement set I indicates that the books were removed from the shelves and placed back in their original order. Consider the mathematical system with the set of elements {R, S, T, U, V, I} with the operation of *.

 To evaluate R * S, write

As shown in Fig. 9.2, R replaces 1 with 3 and S replaces 3 with 3 (no change), so R * S replaces 1 with 3. R replaces 2 with 1 and S replaces 1 with 2, so R * S

Figure 9.2:

replaces 2 with 2 (no change). R replaces 3 with 2 and S replaces 2 with 1, so R * S replaces 3 with 1. R * S replaces 1 with 3, 2 with 2, and 3 with 1.

Since this result is the same as replacement set T, we write R * S = T.

(a) Complete the table for the operation using the procedure outlined.

(b) Is this mathematical system a group? Explain.

(c) Is this mathematical system a commutative group? Explain.

Transcribed Image Text:

(1 2 3 R = \3 2, 1 2 3 2 3 G 3 2) 2 3 U = 3 2 1 3 2 1 2 3 2 3, V = 3 2 3