5. A group < G, o > is a set G together with a binary operation o such that:

• the operation o has closure—for all x, y in G, x o y is in G

• associativity (x o y) o z = x o (y o z)—for all x, y, z in G

• identity exists ∃ e ∈ G such that

∀ x ∈ G x o e = e o = x

• inverses exist—for all x in G there exist x-1 such that x o x-1 = x-1 o x = e Subtraction of natural numbers is not closed as 3 – 7 = −4 and −4 is not in the set of natural numbers. Addition of natural numbers, however, is a binary operation. Addition of integers is associative: (2 + 3) + 4 = 2 + (3 + 4) = 9. However, subtraction of integers is not: (2 − 3) − 4 does not equal 2 − (3 − 4); i.e., –5 does not equal 3; 0 is an identity element for addition of integers 7 + 0 = 0 + 7 = 7. The inverse for 4 with respect to addition of integers is −4 as 4 +
(−4) = 0.
Examples of groups: < Z, +>: The set of integers with respect to addition. : The set of non-zero rationals with respect to multiplication.

a. Consider a square (Sq), shown in Figure 4.28, which is free to move in 3-dimensional space labeled as follows: Let Π0, Π1, Π2 and Π3 be clockwise rotations through 0°, 90°, 180°, and 270°, respectively. And let o represent composition of these rotations. For example Π1 o Π2 is a 90° rotation followed by a 180° rotation which corresponds to Π33 which is a 270° clockwise rotation. Prove that is a group.

b. Apply this group to the tic-tac-toe board introduced in section 4.1. Verify that
gives us justification for the notion of equivalence stated in Figure 4.4. Consult McCoy. 8