Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [boolean-algebra]

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras. For Boolean logic use the tag propositional-calculus.

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras.

A Boolean algebra uses Boolean variables, typically denoted by capital letters, e.g. $A,B$, which can only take the values $0$ or $1$. Operators are $\land$ (conjunction), $\lor$ (disjunction) and $\lnot$ (negation).

For Boolean logic use the tag .

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Karnaugh table verification

I was composing the Karnaugh table for the expression $x'y'z'+x'y'z+x'yz'+x'yz+xy'z+xyz$. The book has the answer: My question is why the last row $1$'s are grouped as a double group whereas taking together above $1$'s could make a quarter and this…
Manjoy Das
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Karnaugh map solution verification

Find the minimal form of the logical expression in its DNF form as \begin{equation*} d=x'y'z'+x'yz'+x'yz+xyz' \end{equation*} My answer: After grouping $(1,3),(1,4)$ and $(1,4),(2,4)$ cells as doubles and $(1,1)$ cell as single, I obtain the…
Manjoy Das
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Help with theorem used in Boolean Expression Simplification

I need to simplify the following expression $$(a+b+\bar c)(a+c+d)(\bar a +b+c) $$ This is what I've done: $$=(a+(b+\bar c)(c+d))(\bar a+b+c)$$ $$=(a+bc+bd+\bar cc+\bar cd)(\bar a+b+c)$$ $$=(a+bc+bd+\bar cd)(\bar a+b+c)$$ $$=a \bar a +ab+ac+\bar…
pumafinger
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How to simplify following the boolean expression $A+\bar{A} B \bar{C}$

Equation to minimize using Boolean Algebra Laws: $A+\bar{A} B \bar{C}$ I have tried doing this but i am unsure of the answer: $$ \begin{array}{l} \text { Let } K=B \bar{C} \\ A+\bar{A} K=A+K=A+B \bar{C} \end{array} $$
Debanjan Dhar
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Simplifying Boolean algebra... What am I missing?

I'm reviewing boolean algebra, but I'm having trouble with a basic simplification: $$\begin{equation}\begin{aligned} &x'z'+ xyz +xz'\\ &= z'(x+x')+xyz\\ &= z'+xyz\\ &= ??? \\ &= z'+xy \end{aligned}\end{equation}\tag{2}\label{eq2}$$ I can't…
LosOnTheGo
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How do I simplify $\overline{X}\overline{Y} + YZ + \overline{X}Y\overline{Z}$?

$$\overline{X}\overline{Y} + YZ + \overline{X}Y\overline{Z}$$ I'm having a lot of trouble simplifying this boolean expression. I used commutative property and re-arranged it as my first step: $$\overline{X}Y\overline{Z} + \overline{X}\overline{Y} +…
S. Coughing
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completeness and saturation

Let $B$ a complete Boolean algebra. Suppose, for $\kappa$ cardinal, that $B$ is not $\kappa$-saturated. Then there exists a partition $W$ of $B$. Because of completeness, we have $B=\sum W\in B$. So $B$ is in an element of an element of $B$ ?
Marc Moretti
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Reduce to lowest possible gates

I am trying to reduce (¬A V ¬B) V (A ⊕ B ) to be expressed by the lowest possible number of gates. So far by expanding the XOR gate and using Demorgan's and distributive laws, I have come down to this ¬(A ∧ B) V (A V B) Is it possible to reduce this…
user15045489
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Simplify the expression: $A'BC' + A'BC + AB'C + ABC' + ABC$

The online calculator(not posting the name because I don't know if its's allowed) is giving me a different result and I can't find what I'm doing wrong: Mine: $A'BC' + A'BC + AB'C + ABC' + ABC= BC'(A'+A) + BC(A' + A) + AB'C= BC' + BC + AB'C= B(C' +…
EL02
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I found two definitions for an atom in a boolean algebra, but cannot find a proof of their equivalence.

I want to show that the following definitions for an atom $a$ are equivalent for a nonzero element $a$ in a Boolean algebra $\mathcal{B}$: for all $x\in\mathcal{B},a\leq x$ or $x\land a=0$ for all $x\in\mathcal{B},x\leq a\Rightarrow x=0$ or…
Sven BandSomething
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Boolean Expression Simplification $A'B'CD + A'B'C'D + ABC'D' + A'BC'D + AB'C'D'$

I want to simplify the below equation but I can't get the same result with an online calculator (the result for the calculator is the correct - I check it with Logisim). $Y = A'B'CD + A'B'C'D + ABC'D' + A'BC'D + AB'C'D'$ $Y = A'B'(C'D + CD) + A'BC'D…
2 X
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Simplify $xyz’w’+xy’zw+xyz’w+xy’z’w’+xy’z’w+zw’xy’$

Simplify $xyz’w’+xy’zw+xyz’w+xy’z’w’+xy’z’w+zw’xy’$ ( ` = not) Using a karnaugh map I found that it's equal to $xy'+xz'$, but I can't find how to get there with simple algebraic simplifications. Help would be appreciated.
paxtibimarce
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Simplify Boolean expression $A'B'C'D' + A'B'CD + A'B'CD' + AB'CD' + AB'C'D'$

Using karnaugh map, I know that my final answer should be $B'D'+A'B'C$ But I cannot simplify this expression to that. So far I got... $A'B'C'D' + A'B'CD + A'B'CD' + AB'CD' + AB'C'D'$ $=A'B'C(D+D')+ A'B'C'D'+ AB'CD' + AB'C'D'$ $= A'B'C + B'C'D'(A'+A)…
TGNDJS
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Show that the Boolean algebra of finite & cofinite subsets of $U$ is not complete

If $U$ is infinite, show that the Boolean algebra of subsets of $U$ that are finite or cofinite (i.e. their complement is finite) is not complete. A Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ is said to be complete if every non-empty…
stoic-santiago
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Preservation of completeness under isomorphisms between Boolean algebras

A Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ is said to be complete if every non-empty subset of $B$ has a greatest lower bound (g.l.b). Prove that a Boolean algebra that is isomorphic to a complete Boolean algebra is complete. Let…
stoic-santiago
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