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Chapter 15: Problem 10

If \(x, y\), and \(z\) are Boolean variables and \(x+y+z=x y z\), prove that \(x, y, z\) all have the same value.

Short Answer

Expert verified
If Boolean variables \(x, y, z\) satisfy the equation \(x + y + z = x y z\), it implies all variables have the same boolean value, either true or false.

Step by step solution

01

Expression Interpretation

The given expression in terms of Boolean algebra is \(x + y + z = x y z\). The '+' operation stands for logical OR and '.' operation stands for logical AND.
02

Consider all possible scenarios

We will proceed by considering all possible cases. Namely, two scenarios: all variables are True, and all variables are False.
03

Case 1: All variables are True

If \(x = y = z = 1\) (where 1 refers to boolean True), the left hand side of the equation \(x + y + z\) becomes 1 (since the result of a boolean OR operation with one true operand is true). Similarly, the right hand side \(x y z\) becomes 1. Hence, the equation holds true which means all variables have the same value.
04

Case 2: All variables are False

If \(x = y = z = 0\) (where 0 refers to boolean False), the left hand side of the equation \(x + y + z\) becomes 0 (since the result of a boolean OR operation with all false operands is false). Similarly, the right hand side \(x y z\) becomes 0. Hence, the equation holds true which means all variables have the same value.
05

Conclusion

From both cases, it's clear that the given equation \(x + y + z = x y z\) will hold true only if all variables (x, y, and z) have the same boolean value, either True or False.

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Most popular questions from this chapter

Let \(x, y\) be elements in the Boolean algebra Prove that \(x=y\) if and only if \(x \bar{y}+\bar{x} y=0\).

Simplify the following Boolean expressions. a) \(x y+(x+y) \bar{z}+y\) b) \(x+y+\overline{(\bar{x}+y+z)}\) c) \(y z+w x+z+[w z(x y+w z)]\) d) \(x_{1}+\bar{x}_{1} x_{2}+\bar{x}_{1} \bar{x}_{2} x_{3}+\bar{x}_{1} \bar{x}_{2} \bar{x}_{3} x_{4}+\cdots\)

Let \(f, g: B^{n} \rightarrow B\). Define the relation " \(\leq "\) on \(F_{n}\), the set of all Boolean functions of \(n\) variables, by \(f \leq g\) if the value of \(g\) is 1 at least whenever the value of \(f\) is \(1 .\) a) Prove that this relation is a partial order on \(F_{n}\). b) Prove that \(f g \leq f\) and \(f \leq f+g\). c) For \(n=2\), draw the Hasse diagram for the 16 functions in \(F_{2}\). Where are the minterms and maxterms located in the diagram? Compare this diagram with that for the power set of \(\\{a, b, c, d\\}\) partially ordered under the subset relation.

Define the closed binary operation \(\odot\) (Exclusive Or) on \(F_{n}\) by \(f \oplus g=f \bar{g}+\bar{f} g\), where \(f, g: B^{n} \rightarrow B\). a) Determine \(f \oplus f, f \oplus \bar{f}, f \oplus 1\), and \(f \oplus 0\). b) Prove or disprove each of the following: i) \(f \oplus g=0 \Rightarrow f=g\) ii) \(f \oplus(g \oplus h)=(f \oplus g) \oplus h\) iii) \(f \oplus g=\bar{f} \oplus \bar{g}\) iv) \(f \oplus g h=(f \oplus g)(f \oplus h)\) v) \(f(g \oplus h)=f g \oplus f h\) vi) \((\overline{f \oplus g})=f \oplus g=f \oplus \bar{g}\) vii) \(f \oplus g=f \oplus h \Rightarrow g=h\)

Find a minimal-sum-of-products representation for a) \(f(w, x, y, z)=\sum m(1,3,5,7,9)+d(10,11,12,13,14,15)\) b) \(f(w, x, y, z)=\sum m(0,5,6,8,13,14)+d(4,9,11)\) c) \(f(v, w, x, y, z)=\sum m(0,2,3,4,5,6,12,19,20,24,28)+d(1,13,16,29,31)\)
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