Exercises

1. Prove that if 𝒙 ⊔ 𝒚 exists then 𝒙 ⊑ 𝒚 ⇔ 𝒙 ⊔ 𝒚 = 𝒚

2. Prove that function composition preserves monotonicity. That is, if 𝒇: 𝑳𝟏 → 𝑳𝟐 and 𝒈: 𝑳𝟐 → 𝑳𝟑 are monotone, then so is their composition 𝒈∘𝒇, which is defined by (𝒈∘𝒇)(𝐱) = 𝒈(𝒇(𝒙)) .

3. Why do these two diagrams not define lattices?

   

4. The operators ⊔ and ⊓ can be viewed as functions. For example, 𝐱𝟏 ⊔ 𝒙𝟐 where 𝒙𝟏 , 𝒙𝟐 ∈ 𝑳 returns an element from L. Show that ⊔ and ⊓ are monotone.

Submission

The exercises should be uploaded to here and named as H3_StuFstName_StuID.pdf, e.g., H2_Kojima_013038.pdf ;