Figure 4: Lattice 2. Show that any complete lattice has a0 and a 3. Prove that a partially ordered set with 1 in which every nonempty set has a g. I.b. is a complete 4. In a bounded distributive lattice, an element can have only one complement 5. Show that in a Boolean algebra the following statements are equivalent for any a and b (a)aUb=b (b)a∩b=a (c)a'Ub=l (d)a∩b=0 (e)a≤b 6. Let A=a, b,c, d, e, f, g, h and R be the relation defined by 11110000 01010000 00110000 00010000 M 01010101 00010001 (a) Show that the poset(A, R)is complemented and give all pairs of complem (b)Prove or disprove that(A, R) is a Boolean algebra. 7. Prove thea b c 1 0 Figure 4: Lattice 2. Show that any complete lattice has a 0 and a 1. 3. Prove that a partially ordered set with 1 in which every nonempty set has a g.l.b. is a complete lattice. 4. In a bounded distributive lattice, an element can have only one complement. 5. Show that in a Boolean algebra the following statements are equivalent for any a and b. (a) a ∪ b = b (b) a ∩ b = a (c) a 0 ∪ b = 1 (d) a ∩ b 0 = 0 (e) a ≤ b 6. Let A = {a, b, c, d, e, f, g, h} and R be the relation defined by MR =             1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1             (a) Show that the poset (A, R) is complemented and give all pairs of complements. (b) Prove or disprove that (A, R) is a Boolean algebra. 7. Prove Theorem 13. 6