With reference to Exercise 14, give a similar alternative labeling for the 8 elements of D4 in Table 8.12. 

Data from Exercise 14

In Table 8.8, we used p_o, p₁ , p₂ , µ₁, µ₂ , µ₃ as the names of the 6 elements of S₃ . Some authors use the notations ∈, p, p² , ∅, p∅, p²∅ for these elements, where their ∈ is our identity p₀ , their p is our p₁, and their ∅ is our µ₁. Verify geometrically that their six expressions do give all of S₃ .

8.12 Table Po PI P2 P3 | fl 8 8 Po Pi P P3 || Po P P2 P P2 f2 P3 Po PI P2 8 8 P P3 Po 8 82 P3 Po Pi H 82 M2 81 PO Po 8 fi 8 fi 1 8 P3 2 8 8 9 P 8 2 f| 8 8 8 H1 | P3 14 3: 82 82 1 8 H2 P2 Po PI P3 P2 P3 P1 P3 Po P2 P P2 Po