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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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Boolean Algebra - proof without associativity?

I would like to prove the following: $(x\cdot y) + (\overline{x} + \overline{y}) = 1$ without the Associativity Property. I can't seem to do this algebraically (without truth tables).
user238211
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simplify the boolean expression $abc|ab\sim c|a\sim bc|\sim abc$

I worked this through to a&c but this has to be wrong. I'm clearly going wrong somewhere. Could someone point out the wrong step in my method? $$(a\land b\land c)\lor (a\land b\land \lnot c)\lor (a\land \lnot b\land c)\lor (\lnot a\land b\land…
freshnewpage
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Boolean Algebra: $a+a'b = a+ab = a$?

$a(a'+b) = aa'+a'b = a'b$ ($aa' = 0$ in any case) $a+a'b = 1a + a(a'+b) = a(1+a'+b) = a$ $a+ab = a(a+b) = a => a+a'b = a+ab$ However, when I use truth table to compare the result of $a+a'b$ to $a+ab$ when $a = 0, b = 1$ $a+ab = 0$ while $a+a'b =…
aukxn
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simplifying boolean expression in maxterm

Should I expand the equation to simplify? Π(1,4,5,6). It means $$ F = (A + B + C')(A' + B + C)(A' + B + C')(A' + B' + C) $$ I have expanded and found $$ = ( C' + AC + A'B)(A' + BC + B'C') $$ I haven't gone on.
AskQuestion
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Consensus Theorem: Legal to use redundant terms to find more redundant terms?

When using the Consensus Theorem in Boolean algebra to minimize an expression, is it a legal move to find and add a redundant term to the expression and then use that term to find more redundant terms and then eliminate all of them? For instance: F…
NDro
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Obtain the Boolean expression from the given circuit diagram

Currently having trouble understanding how to write out the boolean expression up to the exclusive or gate. Up to the third NAND gate I solved it to be AB+CD. But I get stumped on how to write out the fourth gate. Would it be 0 (+) AB+CD then? If…
bevi
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Atoms in a Boolean algebra

I am trying to understand the concept of an atom in a Boolean algebra. To fix the ideas, let $X=\{a,b,c\}$ be a set, and $\mathcal{A}=\{\emptyset,\{a\},\{b,c\},X\}$ be one of the five possible algebras of subsets of $X$. Of course, $\mathcal{A}$…
user60264
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Repeated XOR operations

Suppose you have a list of truth values with $2^k$ elements for any natural number $k$. If the first element of this list is denoted as $L(1)$, then we can come up with a new list by performing the following operation…
cmpen422help
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Boolean simplification with some known term combinations

I am doing boolean simplification using Quine-McCluskey which works well. However, I now need to perform the simplification with some known term combinations. For example, I want to simplify: (A+B)C If I know that: A+B == true then this simplifies…
Chris
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All subalgebras of eight-element Boolean algebra

Let's assume that we have a set: $$ X = \{a, b, c\} $$ Is it true, that a Boolean algebra of this set is like below? $$ P(X) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\} $$ If yes, what are subalgebras of this…
JohnDoe
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Simplifying P AND (P OR NOT Q)

How can I simplify this? I've tried invoking Demorgan's Law and I get P AND (NOT (NOT P AND Q)) but I can't seem to simplify this further. The answer is P, but how can I prove this?
Jason
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Simplifying a boolean expression

Can someone help me simplify this boolean expression? $$(a+b+c+d)(a'+b'+c'+d')$$ so if I use the distributive property, I'll get: $$ab'+ac'+ad'+ba'+bc'+bd'+ca'+cb'+cd'+da'+db'+dc'$$ I'm stuck after this step...
frank
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How many n-ary Boolean functions essentially dependent on each of their arguments?

How many n-ary Boolean functions essentially dependent on each of their arguments? essentially dependent means that $$f(b_1,…,b_{i−1},0,b_{i+1},…,b_n) \neq f(b_1,…,b_{i−1},1,b_{i+1},…,b_n)$$
dehasi
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How to convert a mod 2 function to an expression in Boolean Algebra

I'm not sure if this is the right place to post it but I have a question I'm having a hard time understanding. The questions is: Convert the function $X^3Y + 2XZ + WX + W$ mod $2$ to an expression in Boolean algebra. My professor told us that the…
Anonymous
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Is there any way to simplify the following boolean expression?

I was trying to manipulate with litarals and minterms of this booleans expression but it really did not lead to anything that could simplify the expression further.. Not sure if I am doing it wrong or it indeed cannot be simplified any…
UserMoon
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