ents and an English-language description of what R ◦ R −1 represents. 8.21 Again for the R from Exercise 8.19, what is R −1 ◦ R? Again, give both a list of elements and a description of the meaning....

ents and an English-language description of what R ◦ R −1 represents. 8.21 Again for the R from Exercise 8.19, what is R −1 ◦ R? Again, give both a list of elements and a description of the meaning.

For the following exercises, express the give

R ⊆ Students × Times, where hs,ti ∈ R indicates that student s is taking a class at time.

R ⊆ Rooms × Times, where hr,ti ∈ R indicates that there is a class in room r at time t.

R ⊆ Students × Students, where hs,s ′ i ∈ R indicates that students s and s ′ are taking at least one class in common.

Exercise 8.19

Write down the “is an ingredient of” relation on Ingredients × Sauces using the tabular representation of relations introduced in Figure 8.2