Tagged Questions

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Questions regarding Boolean algebras as structures, or regarding functions defined from/to Boolean algebras fit into this tag very nicely. For Boolean logic use the tag propositional logic

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1answer
13 views

Simplify (x'+y)'(x+y)' with boolean algebra

So I'm doing some homework and trying to simplify (x'+y)'(x+y)'. So far these are the steps I've completed, but I'm not 100% sure that they're appropriate. $(x'+y)'(x+y)' = (x'+y)'(x’y’)$ ...
0
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0answers
9 views

Help with boolean algebra simplification

How can I simplify this using Boolean algebra: wx'z+xy'z I was thinking of distributing the z so it would be z(wx'+xy') but what should I do next? Help will be greatly appreciated.
0
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0answers
8 views

Boolean Algebra simplification

How can I simplify this expression? given: wx'z+y'z'+xz'+xy'z my work: z(wx'+xy')+y'z'+xz'is this step correct? How can I simplify further, maybe to three terms?
0
votes
1answer
17 views

Karnaugh map minimal representation

Find the minimal representation for: f(w,x,y,z)= summation m(0,5,6,8,13.14)+d(4,9,11,12) I was a little confused what to do with the don't cares but I used all of them.. Based on the Karnaugh map ...
3
votes
2answers
45 views

Can this expression $(\neg B \land \neg D) \lor (\neg A \land B \land C) \lor (A \land C \land D)$ be further simplified?

I have assignment for computer architecture where I have to simplify a big boolean function: f(a, b, c, d) = a'b'c'd + a'bcd' + abcd + a'bcd + a'b'cd' + ab'cd' + ab'c'd' + ab'cd + a'b'c'd' ...
0
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0answers
25 views

Show that a interval from a boolean algebra is also a boolean algebra and that a function is surjective

We have an boolean algebra $(B,\lor, \land, ', 0, 1)$ and $b \in B - \{0\}$. We consider $[0,b] = \{x \in B | 0\le x\le b \} \subset B$, where $\le$ means an order relationship introduced in the ...
-1
votes
0answers
6 views

definition of fuzzy translation and fuzzy mutiplication [on hold]

how can a fuzzy subset be a fuzzy translation and fuzzy multiplication in BF/BG-algebra? http://www.indjst.org/index.php/indjst/article/viewFile/37135/29726
0
votes
2answers
14 views

Boolean algebra simplification

Is this the simplest form of the expression? Given: $$x'y'z+x'yz'+x'yz$$ My work: $$x'y(z'+z)+x'yz'= x'y+x'yz= x'y(z)$$
0
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2answers
26 views

how does $(p\to q)\lor r \lor s$ effect $(p\leftrightarrow q) \lor r \oplus s$

If we know that $\lnot p \lor q \lor r \lor s=\top$, then what is the value of: $(\lnot p \land \lnot q) \lor (p \land q) \lor(r \land \lnot s) \lor (\lnot r \land s)$ I tried doing it with a truth ...
-1
votes
1answer
32 views

How to simplify the following expressions to a minimum SOP form using the basic identities of Boolean algebra? [closed]

How to simplify the following expressions to a minimum SOP form using the basic identities of Boolean algebra? $$A'C'+C'D'+AC'+A'CD'+A'B'D+A'BD$$ Thanks
0
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1answer
20 views

Implement using only XOR and AND gates [closed]

How can I implement the function: F = ABC’D + AD’ + A’D using only XOR and AND gates ?
3
votes
0answers
25 views

Proving Boolean Logic using Axioms

I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all ...
1
vote
2answers
32 views

What am I missing about this Boolean expression expansion?

Sorry if this is too basic, but I am working through Boolean Algebra and Its Applications and do not understand this expansion in the author's example 5 in section 1-6: $$(A+X+Y)(A+B'+Y') \rightarrow ...
1
vote
3answers
102 views

Why can't a Venn diagram constitute a proof?

I'm reading Boolean Algebra and Its Applications and come across this statement about Venn diagrams: It should be remembered that such diagrams do not constitute proofs, but rather represent ...
-1
votes
2answers
21 views

Boolean Simplification of a large problem

I am unsure where to even start on this problem. My intuition that what ever can be done to the original problem can be done over and over to simplify the whole thing. Please help with some guidance. ...
1
vote
1answer
18 views

Boolean Simplification of $ (a+b) \cdot (a \cdot c + a \cdot \overline{c}) + a \cdot b + b $

Below is my simplification, but my truth tables don't line up, but I can't find my error. $ (a+b) \cdot (a \cdot c + a \cdot \overline{c}) + a \cdot b + b $ $ (a+b) \cdot a \cdot (c + \overline{c}) ...
1
vote
2answers
36 views

Boolean Simplification of $(\overline{a+b+c})+a\cdot(b+ \overline{c})$

I'm lost, when checking my answer via truth tables, my simplified form does not match the original equation. My work, with reasoning step by step is below. Can you help me figure out where I'm wrong, ...
2
votes
3answers
28 views

What is the most simplified form of $y(x′z + xz′) + x(yz + yz′)$

I am stuck on a problem that I know the logical answer to, yet I cannot seem to simplify properly to get there. I want to simplify $$F(x,y,z)=y(x′z + xz′) + x(yz + yz′)$$ I know the simplest form ...
0
votes
1answer
20 views

Boolean and equivalent to summation

Is there a mathematic symbol to express the application of AND operator to a set of booleans, that returns true only of all booleans in the set are true. Something like the summation operator on a set ...
1
vote
1answer
23 views

If ¬ has a higher precedence than ∨, could one affirm “¬ (p ∨ r) ∨ r” <=> “¬ ((p ∨ r) ∨ r)”?

I'm currently in a disagreement with a colleague over how one should intrepret the precedence of the ¬ operator in boolean algebra, and I hope someone here may enlighten me. We both agree that the ¬ ...
1
vote
0answers
10 views

How should I think when implementing Patrick's method?

I have implemented Quine-McCluskey method of boolean function simplification. I ended up with the table of prime implicants: As you can see my results are the same as these on wikipedia. However ...
0
votes
3answers
40 views

XOR of two numbers AND third number ?

If: x & (a ^ b) != 0 Then one of the following holds: x & a == 0; x & b != 0 or x & b == 0; x & a != 0 What is the reason for this? And are there similar ...
1
vote
0answers
27 views

How do I know that min-term can't be combined any further?

I'm trying to learn (and implement) Quine-McCluskey algorithm for boolean function minimalisation. I'm learning the algorithm from wikipedia example. From that I understood the following: Take all ...
6
votes
1answer
82 views

The ring of idempotents

Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := ...
0
votes
1answer
57 views

Find all the prime implicants for the following Boolean functions, and determine which are essential.

Find all the prime implicants for the following Boolean functions, and determine which are essential: F(W,X,Y,Z) = Im(0,2,5,7,8,10,12,13,14,15) Book solution: Prime = XZ, WX, X'Z', WZ' Essential = ...
0
votes
0answers
21 views

How do you simplify this boolean expression?

Original expression : $$J = ((A′ + B)′ + C′)′ + DC′ + AB′ $$ Here is what I did but the correct answer is $AB' + C + D$. $$J = ((AB') + C')' + DC' + AB'\\ J = ((AB')'C) + DC' + AB' \\ J = ((A' + ...
2
votes
3answers
62 views

Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q.

I have difficulties understanding why $ P \implies Q$ is equivalent to P only if Q. I do understand that in the statement "P only if Q", it means if $ \lnot Q \implies \lnot P$". Regarding this ...
1
vote
1answer
23 views

Simplify this Boolean expression?

X Y Z+X Y' Z'+X' Y' Z+X' Y Z' I know it simplify to (X XOR Y XOR Z),BUT I want to simplified using only AND, OR, and NOT Gates? Please help I spent three hours but I don't get the same truth table.
1
vote
1answer
18 views

Finding Product-of-Maxterms Form

I need help to resolve this problem, i have the following boolean function: [(A.!C)+!(A.!C)].!(A.!B) The Truth table is: (please see this LINK TO wolframalpha ...
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votes
2answers
32 views

Boolean Algebra Simplification

I have to simplify A'BC' + A'B'C + A'BC + ABC My result was A'BC Is this correct?
0
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0answers
5 views

Need to create a specific four variable Boolean function?

I need to create a Boolean function consisting of four inputs (a,b,c,d) that is 1 if no more than 2 of its inputs are 1. I have experienced great difficulty on this particular problem. If anyone ...
2
votes
2answers
44 views

Simplification of boolean expression with xor

I need to simplify the following boolean expression ¬(A xor B) xor (B + ¬C) I know A xor B = ¬AB + A¬B Then the expression will become ¬(¬AB + A¬B) xor (B + ¬C) However, I stuck on it and I don't ...
0
votes
1answer
24 views

Boolean algebra simplification a'bc+ab'c+abc'+abc

Can't figure out how to simplify $(^\neg a)bc+a(^\neg b)c+ab(^\neg c)+abc$, I'm really bad at this...
0
votes
0answers
32 views

Simplify Boolean Function F(w,x,y,z) = wx' + y'z' + w'yz' using Karnaugh map?

Given: F(w,x,y,z) = wx' + y'z' + w'yz' How would I simplify it? Also given the first term wx' would that translate to 10 or would I have to incorporate y and z ...
0
votes
0answers
10 views

Condition on Vector Boolean Function to be Bijective

Suppose the vector boolean function be $$\begin{align} f:F^n_2 \longrightarrow F_2^n \\ (x_1,\dots ,x_n) \longrightarrow (x_2,\dots x_n,g) \\ \\ g:F^n_2 \longrightarrow F_2 \\ (x_1,\dots ,x_n) ...
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0answers
7 views

Relatioship b/w Bent function and Correlation Immunity

I assume that the readers know the definition of Walsh-Hadamart transform for Boolean function. An n-variable Boolean function $f$ is called bent if $W_f(\alpha)=\pm 2^{n/2}$ $\quad \forall \alpha ...
0
votes
2answers
43 views

Equality of two expressions describing a filter

Let $U$, $W$ be boolean lattices with order $\sqsupseteq$, and $U \supseteq W$. The top element of $U$ is the same as the top element of $W$. The bottom element of $U$ is the same as the bottom ...
0
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0answers
22 views

How to simplify a boolean expression with only the algebra laws?

The expression is $$AD^{'}+A^{'}B+C^{'}D+B^{'}C $$ It clearly has some form of symmetry because there are exactly an equal amount of complements for each variable. I have to simplify it to a product ...
1
vote
1answer
51 views

Verify Demorgan's Law Algebraically

If $\overline X \equiv \text { not }X$, De Morgan's Laws are stated as: $ \overline{(A + B)}= \overline A\cdot \overline B$ $ \overline{(A\cdot B)} = \overline A + \overline B$ Verify the above ...
0
votes
1answer
28 views

Demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology.

I'm struggling to demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology. I know that : ...
1
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1answer
28 views

Demonstrate that p ↔ (p ↔ q) ⇔ q

I know the answer is : (p ↔ p) ↔ q ⇔ q 1 ↔ q ⇔ q q ⇔ q But I don't understand why it isn't : ...
0
votes
1answer
13 views

Boolean algebra proof (a+b) (a+c)' = a'bc'

I have to prove that (a+b) (a+c)' = a'bc' My algebra skills are really rusty and I was wondering what identities are used to solve this so I can get a better understanding
0
votes
1answer
18 views

Boolean Algebra - What is the meaning of f(x) = ∑(N)?

Suppose you had: f(x,y,z) = ∑(2,3,4,5) What does this represent or map? For each of the variables x,y,z?
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votes
1answer
15 views

Equivalence of the two boolean expression

This is a question from a textbook on digital logic which I am having a difficult time with: Prove that the following expression is valid: ...
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0answers
36 views

Convert from sum of products to product of sums (Boolean algebra)

I had to simplify a boolean expression with a k-map then put it into a NOR-gate implementation circuit. I haven't made the circuit yet, but here is the work I've done: Original function: $$F(w, x, ...
2
votes
2answers
23 views

Boolean Algebra Minimization

Prove that $\bar{A}B + AC + BC = \bar{A}B + AC$ with the help of boolean algebraic manipulations. I have no clue from where to start, how should I tackle these type of questions? Or $$ ...
0
votes
0answers
21 views

Question on homogeneous algebra

Defn: A Boolean algebra $B$ is homogeneous if for every non-zero $a\in B$, $B$ is isomorphic to $B|_a$. e.g. the algebra $\mathcal L$ of all Lebesgue measurable sets in $[0,1]$ modulo null sets. ...
1
vote
1answer
22 views

Countable additive of a measure

Suppose we have a field of sets $\mathcal F$ such that no infinite union of members of $\mathcal F$ belong to it. Let $m$ be any finitely additive measure on $\mathcal F$, then $m$ is ...
2
votes
2answers
35 views

How would I go from DNF to a simplified formula with less symbols?

Here's a DNF: $$(\neg A_1 \land \neg A_2 \land \neg A_3 ) \lor (A_1 \land \neg A_2 \land \neg A_3 ) \lor (\neg A_1 \land \neg A_2 \land A_3 ) \lor (\neg A_1 \land A_2 \land \neg A_3 )$$ And the ...
1
vote
1answer
20 views

Boolean Algebra: making a proof assistance

So far i've tried all the identities my teacher gave us and keep getting stuck I have to prove that x'y' + y = x' + xy using boolean algebra identities