## 1

Draw the lattice diagram for the power set of $X = \lbrace a, b, c, d \rbrace$ with the set inclusion relation $\subseteq$.

## 2

Let $B$ be the set of positive integers that are divisors of 30. $a \preceq b$ on $B$ is defined by $a \mid b$. Prove that $B$ is a Boolean algebra. Find a set $X$ such that $B$ is isomorphic to $\mathcal{P} (X)$

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## 3

Show whether $\mathbb{Z}^{+}$ with $a \preceq b$ defined by $a \mid b$ is a lattice, and whether it is a Boolean algebra.

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$\mathbb{Z}^{+}$不是布尔代数

## 4

Let $G$ be an Abelian group and $X$ be the set of subgroups of $G$ ordered by $\subseteq$. Prove that the least upper bound of two subgroups $H$ and $K$ in Boolean algebra $X$ is $HK$.

## 5

The order of any finite Boolean algebra must be $2^{n}$ for some positive integer $n$.

## 6

Let $B$ be a Boolean algebra. Prove that

• $a=b$ iif $(a \land b’) \vee (a’ \land b)=O$ for $a,b \in B$
• $a=O$ iif $\forall b \in B, (a \land b’) \vee (a’ \land b)=b$

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