Figure 4: Lattice b)Show that the lattice in shown in the Figure 4 is a nondistributive lattice that is modular 2. how that any complete lattice has a0 and a 1 3. Prove that a partially ordered set with 1 in which every nonempty set has a g.1.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)aUb=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 1110000 01010000 00110000 00010000 M 01010101 00110011 (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 1a b c 1 0 Figure 4: Lattice (b) Show that the lattice in shown in the Figure 4 is a nondistributive lattice that is modular. 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 ′ ∪ b = 1 (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 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