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 .

3083 questions
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Simplifying boolean expression: $!(x!z+y!z+xy+z)$

This is the expression: ', ! not+ or $((x'y'+z)'+z+xy+wz)'$ After some steps I can get $!(x!z+y!z+xy+z)$ How can I continue from here?
mjumurs
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$(p\land \lnot q) \lor q \lor (\lnot p\land q)$

The Boolean Expression $(p\land \lnot q) \lor q \lor (\lnot p\land q)$ is equivalent to: $(1) p \land q \space \space (2) p \lor q \space \space (3) p \lor \lnot q\space \space (4) \lnot p \land q $ My attempt: $pq'+q+p'q = pq' + q$.…
aarbee
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Boolean Algebra: Simplefication, $(AC + BD)(AC)'(BD)' = (AC)(AC)' + (BD)(BD)'$

I want to know, how to prove this, $(AC + BD)(AC)'(BD)' = (AC)(AC)' + (BD)(BD)'.$ Please, help me.... Thanks. What laws in that line?
user768379
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Eliminating the number of variables in an underdetermined system of Boolean equations

As an example, in an underdetermined system of linear equations, we can eliminate the number of variables by substituting them. As an example: a+b+c+d+e=5 a+2b+3c+4d+5e=13 a+5b+4c+2d+9e=22 With five variables and three equations, we can reduce the…
Biology nerd
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simplify the boolean expression [ (xy')' * (x'+y') ]'

Simplify the boolean expression $(xy')\cdot(x'+ y')'$. Here is my work: $(xy')\cdot(x'+ y')' = [(x' + y'') · (x' + y')]'$ de Morgans rule (b) $= [(x' + y) · (x' + y')]'$ involution $= (x' + y)' + (x' + y')'$ de Morgans rule (b) $= (x''·y') +…
DanS
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Boolean simplification question - $\neg A\neg B + \neg A\neg C + \neg A\neg BC$

My question is how do I reduce the Boolean simplification question - $\neg A\neg B + \neg A\neg C + \neg A\neg BC$ to get $\neg A+BC$. I'm so lost just been trying to get it for awhile only using the 10 boolean simplification rules.
Saisiot
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Simplify F = y'z' + xy'z + xz' + x'yz'

I'm taking Computer Architecture and we were given a list of simple Boolean Rules. I'm having trouble finding out where to start. This is my work so far F = y'z' + xy'z + xz' + x'yz' F = y'((1)z'+x(1)z)+z'(x+x'y(1)) F = y'(z'+xz)+z'(x+x'y) F =…
Zoe Brig
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Simplify Boolean Expression Condensing

I need help proving the following: Show that…
Alireza Firouzja
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Simplify the following Boolean phrases by using the Boolean basic algebra statements

Simplify the following Boolean phrases by using the Boolean basic algebra statements: $(ac)+(bc)+(!ab)$ to the expression $(ac)+(!ab)$.
salah112
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Complement to a product of boolean variables

Consider set of boolean functions of 5 variables: $x_1, x_2, x_3, x_4, x_5$. Let $p = x_1x_3$ and define complement of $p$ as follows: $c(p) = x_2x_4x_5$. Is it possible to construct a general formula which would provide a complement for arbitrary…
Ordev Agens
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Prove the identity: $\neg((a\wedge b)\vee(a\wedge\neg c)) = (\neg a\wedge b)\vee(\neg b\wedge\neg c)$

I've been trying to prove this, but I have no clue how to take it to the end. I can go up to the part: $$\neg a\wedge\neg b\wedge (c\vee\neg c)\vee\neg a\wedge c\vee\neg b \wedge\neg c$$ After this, I do not understand how to get rid of $\neg…
Riona Sopi
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Correct way to simplify $x+xy$ and $xy+xz+x'z+zx$

For Boolean variables, which of the following is correct? $$x+xy= x(1+y)= xy,$$ or $$x+xy= (x+y)(x+x)= (x+y)x?$$ And as a second problem, $$xy+xz+x'z+zx = xy+z(x+x')+zx= xy+z(1+x)= xy+zx.$$ Does it end here or can it be simplified further?
Sakib Arifin
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Maximum number of prime impliacants for conditional statements in a Karnaugh map with n variables?

I'm looking to find the maximum number of prime implicants for a conditional statement in a Karnaugh map with n variables. Example: A is a variable with a domain of {0, 1, ..., 15} and I have a condition A > 5. Here I will have the following…
Elahe
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Logic formula satisfied when number of positive boolean-variables in a list is even

Having a list of boolean variables $(x_1, x_2, ..., x_n)$ I wonder if it would be possible to create a boolean logical formula (using and, or, not) that returns true if the number of true values in the variable list is even? Example $P([0,0,1,1,1])…
Codicef9
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Simplify into sum of products

Simplifying expression into SOP or POS expressions How do i go from 5,9,13 to p,q,r,s? I have the answer but unsure how to derive it. Answer: G(p,q,r,s) =  M(5, 9, 13) = (p+q'+r+s')∙(p' + q + r + s')∙(p'+q'+r+s') [distributive] = ( ( q' + r…
Mason
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