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Chapter 15: Problem 6

Let \((\mathscr{B},+, \cdot,-, 0,1)\) be a Boolean algebra that is partially ordered \(\mathrm{by} \leq .\) If \(w, x, y, z \in \mathbb{B}\) with \(w \leq x\) and \(y \leq z\), prove that (a) \(w y \leq x y\); and (b) \(w+y \leq x+z\)

Short Answer

Expert verified
By observing the rules of Boolean algebra and order relations, it's possible to prove that if \(w \leq x\) and \(y \leq z\) in an ordered Boolean algebra \(\mathscr{B}\), then (a) \(wy \leq xy\) and (b) \(w+y \leq x+z\).

Step by step solution

01

Review the ordered \(\mathscr{B}\) and the premise

Begin by reviewing ordered Boolean algebra \((\mathscr{B},+, \cdot,-, 0,1)\). It's given that \(w \leq x\) and \(y \leq z\). This means that for any \(w\) and \(x\), \(w\) is less than or equal to \(x\), and for any \(y\) and \(z\), \(y\) is less than or equal to \(z\).
02

Prove \(w y \leq x y\)

Start by assuming \(w \leq x\). In a Boolean Algebra, \(a b \leq a c\) if \(a = 0\) or \(b \leq c\) for any \(a, b, c \in \mathscr{B}\). In this case,if \(w = 0\), then \(w y = 0 \cdot y = 0\), and \(0 \leq x y\) for any \(x, y \in \mathscr{B}\). If \(w \neq 0\), then \(y \leq y\) and \(w y \neq w \cdot 1 = w\). Thus, \(w y \leq x y\) for both cases.
03

Prove \(w+y \leq x+z\)

Now, assume \(w \leq x\) and \(y \leq z\). In Boolean algebra, \(a + b \leq c + d\) if \(a \leq c\) and \(b \leq d\) for any \(a, b, c, d \in \mathscr{B}\). It then follows that \(w+y \leq x+z\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partially Ordered Set
In the realm of mathematics, a partially ordered set is a collection of elements where not every pair of elements can necessarily be compared, but for those that can, the relationship maintains certain properties. Specifically, for any elements a, b, and c within such a set, the following conditions hold:
  • If a 鈮� a (reflexivity),
  • If a 鈮� b and b 鈮� a, then a = b (antisymmetry),
  • If a 鈮� b and b 鈮� c, then a 鈮� c (transitivity).

These properties are vital to understanding how elements relate within the structure. In the context of the given exercise, a Boolean algebra forms a partially ordered set through the ordering relation 鈮�, which naturally aligns with Boolean operations. If we know that w 鈮� x and y 鈮� z, we have information about how these elements relate to one another, laying the ground for further algebraic manipulations and proofs.
Boolean Operations
Boolean algebra revolves around Boolean operations, which are fundamental to the field of computer science and digital logic design. The primary operations in Boolean algebra are AND (denoted as 鈰�), OR (+), and NOT (denoted as 鈭� or bar over the variable). These operations abide by specific laws that resemble conventional algebra but under binary values of 0 and 1, where 0 typically represents false, and 1 represents true.

For instance, to prove wy 鈮� xy as in the exercise, the properties of the AND operation (鈰�) in a Boolean algebra are utilized. If w = 0, wy automatically becomes 0, which is the least element in Boolean algebra; hence wy 鈮� xy holds. If w 鈮� 0, the given that y 鈮� z allows us to infer wy 鈮� wy, and by the properties of the set, eventually, wy 鈮� xy.
Algebraic Proofs
The process of validating the truth of a mathematical statement using algebraic principles and logical deductions is known as an algebraic proof. Proofs are fundamental to mathematics as they provide a systematic way to demonstrate without a doubt that a statement is true. Algebraic proofs in Boolean algebra involve utilizing the algebra's axioms, theorems, and properties to show that certain equations and inequalities hold under every circumstance.

Within the scope of our exercise, we've engaged in a basic type of algebraic proof. We leveraged the ordered nature of a Boolean algebra set and the properties of Boolean operations to show that wy 鈮� xy and w+y 鈮� x+z. This approach involved assuming the premise to be true and then logically constructing a pathway to reach the required conclusion, ensuring that each step is justified based on established algebraic rules.

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Most popular questions from this chapter

Eileen is having a party and finds herself confronted with decisions about inviting five of her friends. a) If she invites Margaret, she must also invite Joan. b) If Kathleen is invited, Nettie and Margaret must also be invited. c) She can invite Cathy or Joan, but definitely not both of them. d) Neither Cathy nor Nettie will show up if the other is not invited. e) Either Kathleen or Nettie or both must be invited. Determine which subsets of these five friends Eileen can invite to her party and still satisfy conditions (a) through (e).

Find a minimal-sum-of-products representation for a) \(f(w, x, y)=\sum m(1,2,5,6)\) b) \(f(w, x, y)=\prod M(0,1,4,5)\) c) \(f(w, x, y, z)=\sum m(0,2,5,7,8,10,13,15)\) d) \(f(w, x, y, z)=\sum m(5,6,8,11,12,13,14,15)\) e) \(f(w, x, y, z)=\sum m(7,9,10,11,14,15)\) f) \(f(v, w, x, y, z)=\) $$ \sum m(1,2,3,4,10,17,18,19,22,23,27,28,30,31) $$

If \(g: B^{7} \rightarrow B\) is a Boolean function of the Boolean variables \(x_{1}, x_{2}, \ldots, x_{7}\), how many 1 's are needed in the Karnaugh map of \(g\) in order to represent the product term (a) \(x_{1} ;\) (b) \(x_{1} x_{2}\) (c) \(x_{1} \bar{x}_{2} x_{3} ;\) (d) \(x_{1} x_{3} x_{5} x_{7} ?\)

Let \(\mathscr{B}_{1}\) be the Boolean algebra of all positive integer divisors of 2310 , with \(\mathscr{B}_{2}\) the Boolean algebra of all subsets of \([a, b, c, d, e\\}\) a) Define \(f: \mathscr{B}_{1} \rightarrow \mathscr{B}_{2}\) so that \(f(2)=\\{a\\}, f(3)=\\{b\\}\), \(f(5)=\\{c\\}, f(7)=\\{d\\}, f(11)=\\{e\\} .\) For \(f\) to be an isomorphism, what must the images of \(35,110,210\), and 330 be? b) How many different isomorphisms can one define between \(\mathscr{B}_{1}\) and \(\mathscr{A}_{2} ?\)

For (a) \(n=60\), and (b) \(n=120\), explain why the positive integer divisors of \(n\) do not yield a Boolean algebra. (Here) \(x+y=\operatorname{lcm}(x, y), x y=\operatorname{gcd}(x, y), \bar{x}=n / x, 1\) is the zero element, and \(n\) is the one element.)
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