2. For every a E B, a is a complement of a' according to definition, so(a)=a. It means that ′ is a bijection. 3.Leta≤b,a∩b≤bnb=0.So, we have bn1=bn(aUa)=(bna)u(na)=bna, hich implies b< a. It also means 'inverts the order. Given a,b,letd=aUb, we have a,b≤ d and d≤a,b. Then we have a"≤a∩b. But for anye≤a'∩b, we have el≤a, b and a,b≤e, which imply e≤aUb=d.Soe'≤ d and it means(aUb)=(a′nb) Similarly, we can also prove(a∩b)=a∪b. It is proved The technique used in this proof is the same as the one used in the last theorem of the previous lecture 4.1 Ring and Boolean algebra Finally, we will establish a connection between a ring and a Boolean algebra. Actually a boolean algebra is equivalent to a special class of ring Given a Boolean algebra B, we first show you how to derive a ring. The point is to define two operations and. of a ring. We define them by n and U as the following 1. Addition: a+b=(anb)u(a'nb), it is also called the symmetric difference of a and b 2. Multiplication:a·b=a∩b. The remaining task is to check these two operations satisfying all laws required by a ring, which is tedious and left as an exercise. Then a boolean algebra introduces a ring. Furthermore it is a special ring with a+a=0 and aa=a Conversely, a special ring can also introduce a Boolean algebra. We first introduce a concept Definition 12. A ring is called Boolean if all of its elements are idempotent Given a Boolean ring B with an identity, we just define aUb=a+b-ab and anb= ab. It is easy to check that(B, U, n) is a Boolean algebra Then we have the following theorem Theorem 13. A Boolean algebra is equivalent to a Boolean ring with identity Exercise 1. A lattice is said to be modular if, for all a, b, c, a s c implies that aU(bnc=(aubnc (a) Show that a distributive lattice is modular2. For every a ∈ B, a is a complement of a ′ according to definition, so (a ′ ) ′ = a. It means that ′ is a bijection. 3. Let a ≤ b, a ∩ b ′ ≤ b ∩ b ′ = 0. So, we have b ′ ∩ 1 = b ′ ∩ (a ∪ a ′ ) = (b ′ ∩ a) ∪ (b ′ ∩ a ′ ) = b ′ ∩ a ′ , which implies b ′ ≤ a ′ . It also means ′ inverts the order. Given a, b, let d = a ∪ b, we have a, b ≤ d and d ′ ≤ a ′ , b′ . Then we have d ′ ≤ a ′ ∩ b ′ . But for any e ′ ≤ a ′ ∩ b ′ , we have e ′ ≤ a ′ , b′ and a, b ≤ e, which imply e ≤ a ∪ b = d. So e ′ ≤ d ′ and it means (a ∪ b) ′ = (a ′ ∩ b ′ ). Similarly, we can also prove (a ∩ b) ′ = a ′ ∪ b ′ . It is proved. The technique used in this proof is the same as the one used in the last theorem of the previous lecture. 4.1 Ring and Boolean Algebra Finally, we will establish a connection between a ring and a Boolean algebra. Actually a Boolean algebra is equivalent to a special class of ring. Given a Boolean algebra B, we first show you how to derive a ring. The point is to define two operations + and · of a ring. We define them by ∩ and ∪ as the following: 1. Addition: a + b = (a ∩ b ′ ) ∪ (a ′ ∩ b), it is also called the symmetric difference of a and b. 2. Multiplication: a · b = a ∩ b. The remaining task is to check these two operations satisfying all laws required by a ring, which is tedious and left as an exercise. Then a Boolean algebra introduces a ring. Furthermore, it is a special ring with a + a = 0 and a · a = a. Conversely, a special ring can also introduce a Boolean algebra. We first introduce a concept. Definition 12. A ring is called Boolean if all of its elements are idempotent. Given a Boolean ring B with an identity, we just define a ∪ b = a + b − ab and a ∩ b = ab. It is easy to check that ⟨B, ∪, ∩⟩ is a Boolean algebra. Then we have the following theorem. Theorem 13. A Boolean algebra is equivalent to a Boolean ring with identity. Exercise 1. A lattice is said to be modular if, for all a, b, c, a ≤ c implies that a ∪ (b ∩ c) = (a ∪ b) ∩ c. (a) Show that a distributive lattice is modular. 5