3
votes
1answer
44 views

Simplify a boolean algebra expression: xy + xz' + x'yz

I need to simplify xy + xz' + x'yz into xz' + yz. I know that these expressions are equal in truth value, but I'm not sure how ...
1
vote
3answers
45 views

Functions for boolean operators, that return 1 or 0

Are there any purely mathematical expressions that are equivalent to boolean operators and return $1$ or $0$? For example: $a > b$ Is there any $f(a, b)$ for which if $a>b$, $f(a,b)=1$ and if ...
0
votes
0answers
18 views

The empirically-obvious statement about minimization of Boolean functions

The statement: $\forall f,g: \{0;1\}^n \to \{0;1\} \; (n > 0),$ if $$|f^{-1}(1)| > |g^{-1}(1)|$$ then $f$ has the (non-strictly-)simpler minimization than $g$. $\text{ }$ As mentioned, the ...
0
votes
1answer
30 views

Finding a simple function for a given Karnaugh diagram

I came across one question in which the Karnaugh map for some function is given and using it, i have to find a simple function which gets mapped onto that map. My Attempt: Corresponding to every ...
2
votes
1answer
17 views

Need help proving that $ fRg \Leftrightarrow fg = f $ on $ B^{n} $ to $ B $ if and only if $ f(b_1,…,b_n) \leq g(b_1,…,b_n) $

I'm trying to gather my thoughts for proving the following claim: For $ fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $ fRg $ if and only if for any input values $ b_1,...,b_n $, we ...
0
votes
2answers
60 views

Boolean algebra. For all x, y, and z in B, if x + y = x + z and x × y = x × z, then y = z.

In the statements below, $B$ is a Boolean algebra with $\times$ and $+$ for binary operations and ($\bar{a}$)is the complement of $a$. 4.) For all $x$, $y$, and $z$ in $B$, if $x + y = x + z$ and $x ...
1
vote
1answer
30 views

For all $a$ and $b$ in $B$, $(a \times b) + a = a$.

In the statements below, $B$ is a boolean algebra with $×$ and $+$ for binary operations. 3.) For all $a$ and $b$ in $B$, $(a ×b) + a = a$. This is what I have as an answer. Can someone confirm ...
0
votes
1answer
28 views

Why is this a boolean algebra

Let $A = \{a,b\}$. The $\mathcal P(A) = \{\emptyset,\{a\},\{b\},A\}$. Let $+$ be $\cup$, $\cdot$ be $\cap$, complement be set complement, $1$ be $A$, and $0$ be $\emptyset$. I need to explain why ...
1
vote
1answer
53 views

What is the number of self dual boolean functions?

The dual of a Boolean function $F(x_1,x_2 \dots x_n,+,\bullet)$, written as $F^D$, is the same expression as that of $F$ with $+$ and $\bullet$ swapped. $F$ is said to be self-dual if $F=F^D$. What is ...
0
votes
0answers
22 views

Simplifying conjunctive normal form

I have to simplify the following conjunctive normal form of a polynomial : $(x_1'+x_2+x_3+x_4)(x_1'+x_2+x_3+x_4')(x_1'+x_2'+x_3+x_4')$ I started off by using the fact that$(\alpha + \beta)(\alpha + ...
0
votes
1answer
32 views

Finding conjunctive normal form of a Boolean polynomial

I have to find the conjunctive normal form of the following Boolean Polynomial : $(x_1+x_2+x_3)(x_1x_2+x_1'x_3)'$ I simplified this polynomial to get $x_1x_2'+x_1x_2x_3'$ for which i then formed the ...
1
vote
1answer
24 views

Simplification of a boolean polynomial

I have to simplify the following Boolean polynomial using $x\land y$ = $xy$ and $x\lor y$ = x+y : $xy'+x(yz)'+z$ =$xy'+x(y'+z')+z$ =$xy'+xy'+xz'+z$ =$xy'+xz'+z$ My book gives the following ...
0
votes
2answers
50 views

finding a boolean function with specific property

The problem I am trying to solve is: Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$. My solution is $$\left(p\wedge\thicksim ...
1
vote
2answers
58 views

How to prove the divisors of 15 form a Boolean algebra

This from Exercise 3.1 in "A Beginner's Guide to Discrete Mathematics" Let B be the set of all positive integer divisors of 15, that is B = {1, 3, 5, 15}. Prove that B forms a Boolean algebra with ...
2
votes
1answer
54 views

Is this a good enough proof?

Is this proof good enough? If not, any feedback would be appreciated. Thanks. Either exhibit $333 $ different boolean functions on the three variables $p; q; r,$ or prove that there aren’t $333$ ...
0
votes
0answers
22 views

Simple explanation of a Boolean function?

I took on the challenge to self study discrete math and I've come to Boolean functions. Please note, that I'm new to set notation (just learned it) and the form of the Boolean function confuses me ...
0
votes
1answer
58 views

Boolean Queries in First Order Logic

I understand first order logic and how its constructed but I have some trouble understanding how the following statement and its FO query are formed. This is from a book. ...
1
vote
1answer
44 views

A problem with a boolean function.

This is one of my assignment problem and I almost finished others. This question confused me for quite a long time and now I get nothing about that. I did understand what is an increasing function ...
1
vote
1answer
291 views

Examples of boolean functions in conjunctive normal form

Give an $n$-variable Boolean function $f(x_1; x_2; \cdots ; x_n)$ in conjunctive normal form so that $f$ is $1$, respectively, (a) If at least one of the $n$ variables is $1$; (b) If at most one of ...
0
votes
2answers
47 views

Boolean Logic - Why doesn't the Associativity Law work in this scenario?

By the associativity law, shouldn't the statement below be true? I understand that the truth tables are different but where exactly does the associativity law apply then if this is False? ...
1
vote
2answers
145 views

Boolean Simplification (ABCD)' + ((CD)'(B+D)'

I have to simplify (ABCD)' + ((CD)'(B+D)' function using boolean algebra. I simplified it using a truth table and got ...
1
vote
1answer
87 views

Simplification of a Boolean Expression

I want to simplify this expression: ACD' + E(A+C)(A'+D') + A'C . The result must be a product of sums, where every sum should be consisted of just two variables. For example (A+B)(C+A)(Z+Y) ... ...
0
votes
1answer
96 views

Zhegalkin polynomial Boolean algebra

I have to find the Zhegalkin polynomial of $ (x\rightarrow y)\rightarrow z $. Please tell me if this is right: my polynomial is of this kind $ a_{0} + a_{1}x + a_{2}y + a_{3}z + a_{4}xy + a_{5}yz + ...
0
votes
2answers
425 views

DNF and CNF logic problem

So i want to find the DNF and CNF of : $ x \oplus y \oplus z $ . I tried by using $ x \oplus y = (\neg x\wedge y) \vee (x\wedge \neg y) $ but it got all messy and stuff, I also plotted it in ...
0
votes
1answer
91 views

Logic boolean algebra problem

so I have to prove that these equations : are equivalent?
1
vote
0answers
54 views

Expansion of subsets of a hamming ball in hypercube

Consider a hypercube graph $G_n = (V,E)$ in n dimensions. Let $H_{1/2} \subset V$ be the set which represents the hamming ball of radius $n/2$. That is for every $v \in H_{1/2}$ the hamming weight of ...
0
votes
1answer
34 views

Write $p \rightarrow \lnot q$ in CNF form with only and ,or and brackets

Write $p \rightarrow \lnot q$ in CNF form with only and, or, and/or brackets How on earth would I even do this? Completely lost! Any help appreciated.
0
votes
0answers
87 views

Find the disjunctive normal form of a function

so I am following in the textbook and have just been able to determine the disjunctive normal form of a function given a chart, but now these new questions say: ...
0
votes
0answers
47 views

Definition of a linear extension (total order?) of a poset

Hey I have a question about the definition of a linear extension of a poset. If I was given a hasse diagram of a poset with relation <= (S, <=), how can I get the compatible total order of this ...
0
votes
0answers
68 views

Hasse diagram vs digraph and bounds question

I have attached a link to my hasse diagram I drew... Sorry about image size and rotation. So is it correct to say a hasse diagram is just a digraph with each internal vertice removed? So would my ...
0
votes
1answer
100 views

How to prove this equality using boolean algebra?

I have approximately no idea on how to solve the following problem, so any help would very much be appreciated: $$x' y'+ x y = (x y' + x' y)'$$ I can't figure out how to prove the equalities ...
0
votes
1answer
155 views

Find the disjunctive normal form and then simplify

Let $f(x,y,z,w)=zw+z'w'+xy'z'w+xyz'w$ Disjunctive normal form $zw(x'+x)(y'+y)+z'w'(x'+x)(y'+y)+xy'z'w+xyz'w=(zwx'+zwx)(y'+y)+(z'w'x'+z'w'x)(y+y')+xy'z'w+xyz'w$ ...
0
votes
1answer
38 views

In a boolean matrix, what does the $n$ in $M_{R^n}$ represent?

I'm now learning about binary relations. I stumbled upon this question in the book: Given $A = \{1,3,5,6\}$ and $R$ is a relation over $A$, whose matrix is defined by $$\begin{pmatrix} 0 ...
1
vote
1answer
135 views

Minimization of boolean function using Quine–McCluskey algorithm

I have a boolean row. It looked like this: Y = 0,1,0,1,1,0,0,1,1,0,1,0,1,1,0,0 Then I converted it to: f(x1,x2,x3,x4) = 0101 ∪ 1001 ∪ 1010 ∪ 1100 I divided it into groups: 0 | - ...
1
vote
1answer
342 views

Boolean formula over 64 Boolean variables X

This question comes from this homework assignment from ECS20 at UC Davis. Chess is played on an 8 x 8 board. A knight placed on one square can move to any unoccupied square that is at a distance ...
2
votes
2answers
361 views

Disjunctive normal form expansion

I do not understand this at all. Find the sum-of-products expansions of these Boolean functions. $F(x, y, z) = x + y + z$ $F(x, y, z) = (x + z)y$ $F(x, y, z) = x$ $F(x, y, z) = x y$ ...
2
votes
2answers
211 views

Parity function proofing for every n>=1 using only AND, OR, 0, and 1

Consider the parity function: $F_n$($x_1$, $...$ ,$x_n$) $=$ $\oplus_{i=1}^n$$x_i$ where each $x_i$ is boolean. Prove that, for every $n \ge 1$, there is no way to compute $F_n$ using only ...
0
votes
2answers
364 views

Writing a boolean formula and logic circuit that computes mux

Let $mux(p_{11}, p_{10}, p_{01}, p_{00}, x_1, x_0) = P_{x1x0}$ (with all variables bits). Write a boolean formula, and then draw a circuit, that computes mux. For ...
2
votes
2answers
202 views

Steps to simplify a Boolean Expression

Simplify: (x ∧ y) ∨ (x ∧ ¬y) ∨ (¬x ∧ y) I need to simplify this using the using properties going step by step. I keep ending up with (x ∧ y) as the answer but when I map is out I get that is should ...
0
votes
1answer
425 views

Conclude the premise using rules of inference

First question I have solved I belive... show, s -> (q -> r) <-> (s ^ q) -> r using the defintion of implication and Boolean algebra. s ->(~q V r) <-> ~ (s ^ q ) V r ~s V (~q V r ) <-> ~s ...
1
vote
1answer
129 views

Expressing boolean functions using the not or operator

I need to express these with $\downarrow$ $x+ y + z$ This one I think I can do, I guess at it and copy the wikipedia page since my book has no explanation on how to do this I get $(x \downarrow y ...
0
votes
1answer
119 views

Prove that S is a Boolean Algebra

Let $n\ge1\in\Bbb N$, we define the set of binary boolean vectors with $\varphi^n .$ Prove that $\varphi^n$ is a boolean algebra. So (...) I know that: Let $\varphi=\{0,1\}, \mathrm ...
1
vote
3answers
391 views

Is “(p AND q) OR r” logically equivalent to “p AND (q OR r)” ??

In the context of discreet math / boolean algebra / logic, is "(p AND q) OR r" logically equivalent to "p AND (q OR r)"? I believe so, but my professor said: ...
0
votes
2answers
110 views

Rewrite equivalent boolean function for p ⇔ q

Using only the operators ⇒ (conditional) and ∼ (negation) Rewrite p ⇔ q How should I go about this? Thanks
1
vote
1answer
165 views

Converting into CNF Form

If you have disjunctive clause comprising of n literals for example $(X_1\cup X_2\cup X_3\cup\cdots \cup X_n)$. where $n\geq 4$. How you can convert it into CNF (Conjunctive Normal Form) of $n-2$ ...
2
votes
1answer
89 views

How to find the minimum expression(s) of a set of fixed-width bit fields?

If we define $x_1 x_2 \cdots x_n$ as a bit field of width $n$, and each element $x_i$ may be $0$, $1$, or wildcard $*$. A set of 4-width bit fields $\{0000, 0001, 0100, 0101\}$ can be aggregated ...
2
votes
2answers
93 views

Boolean Algebra-Simplification Assistance Needed

I have to show that (!(P.Q) + R)(!Q + P.!R) => !Q by simplifying it using De Morgan's Laws. Here is what I did but I'm not sure it's right. (!(P.Q) + R)(!Q + P.!R) => !Q (!P + !Q + R)(!Q + P.!R) ...
1
vote
0answers
235 views

Understanding Sum of product and complete sum of product

I have a pair of problems, the first two of my homework, and I'm already unclear on how finding SOP and CSOP for them work. The first: E=xy(1+z)y' It seems like this just reduces to 0, since 1+z ...
2
votes
2answers
913 views

How to apply De Morgan's law?

If for De Morgan's Laws $$( xy'+yz')' = (x'+y)(y'+z)$$ Then what if I add more terms to the expression ... $$(ab'+ac+a'c')' = (a'+b)(a'+c')(a+c)?$$
3
votes
2answers
324 views

Reducing Boolean expressions

Just learning mathematical proof writing and came upon this interesting question Writing an expression using logic. $$(P \land Q \land \lnot R) \lor (P \land \lnot Q \land \lnot R) \lor (\lnot P ...