1
vote
2answers
42 views

Finding the contrapositive of the statement “I go to school if it does not rain”

I got this question in a exam.There were two more statements in the examination(but they were quite clearly wrong).However I got stuck between these two statements.The contrapositive of the the ...
2
votes
3answers
46 views

Do brackets around negation signify negating the input or output - Boolean Algebra Logic Circuits

I know that $\overline{p + q}$ will result in the input to the logic gate being p, and q, and we can negate this by using an or gate, followed by a not gate, or we can just use a nor gate. However, ...
0
votes
0answers
25 views

Dont understand application of Demorgans Law when simplifying to Sum of Products Form - Boolean Algebra

I'm having trouble trying to understand how this can be simplified. I'm stuck at the first part... The expression is: where a AND b is ab or a*b, and ...
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
53 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 ...
3
votes
1answer
73 views

Prove if Tautology, Contradicton, or Neither. Is my proof ok?

Determine whether $((p \Rightarrow q) \Rightarrow r) \Leftrightarrow (p \Rightarrow (q \Rightarrow r))$ is a tautology, a contradiction, or neither. If $p,q,r = (0,0,0)$ then $((p \Rightarrow ...
2
votes
1answer
53 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 $333$ ...
1
vote
0answers
11 views

Logic subject-reductio ad absurdum

Can you solve this using method reductio ad absurdum? 1)A ↔ (¬ B v C) ¬ A ¬ B 2)¬(R∧ (S v T)) 3)R∧¬ T S ¬R∧ S
0
votes
3answers
76 views

Discrete Mathematics (boolean)

Either exhibit 333 different boolean functions on the three variables p; q; r, or prove that there aren’t 333 different such functions $p$ $q$ $r$ $0 0 0$ $001$ $010$ ...
0
votes
1answer
59 views

Checking Boolean Algebra work - Simplification

I am currently working on an assignment for a CE class I am taking, and I wanted to know if I have been simplifying these equations correctly. I'm supposed to reduce them to a sum of products. 1) ...
0
votes
2answers
193 views

Every boolean function is constructed from $\wedge$'s and $\vee$'s

Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$. Here is my solution, can I ask for a feed back on my solution please? $p∧q$ ...
0
votes
0answers
11 views

Cardinality of a subalgebra of a boolean algebra

Let $X$ be a subset of a Boolean algebra $B$, and let ,$A$ be the subalgebra generated by $X$. Show that, if $X$ is finite, then $|A|$ $\leq$ $2^{2^{|X|}}$ , and that, if X is infinite, then |A| = ...
0
votes
1answer
47 views

Discrete Mathmematics

Are the boolean functions $(p\wedge \neg q)\vee (\neg r\wedge q)$ and $(p\vee \neg q)\wedge (r \vee \neg q)$ equal? Explain your answer. Here my solution, Please give me a feed back on this solution, ...
0
votes
1answer
26 views

Boolean Functions-Algebraic rules for Boolean functions-Associative Rule

Is the function $(p \wedge q) \vee r$ equal to the function $p \wedge (q \vee r)$? Let $a(p,q,r)=(p \wedge q) \vee r$ $b(p,q,r)= p \wedge (q \vee r)$ By associate law $a=b$, but using $a(0,0,1)=1$ and ...
0
votes
1answer
102 views

Two question on ternary Cantor set & Jordan content

Is it true that all subsets of the Cantor set have Jordan content zero? What is the definition of countably generated Boolean algebra? Does the Boolean algebra of subsets $[0,1]$ which ...
2
votes
2answers
142 views

Is my answer for this truth table & boolean expression correct?

I was given the following boolean diagram: I had to come out with the truth table and the simplified expression. I need help to check if my answers are correct below.
1
vote
1answer
54 views

boolean algebra simplyfing

I need to solve these expressions with boolean algebra: $$Z = a'bc+ab'c+abc'+abc\mbox{ AND }Z = a'(b+db'c')+a'b'(d'c'+c)$$ Every advice is more then welcome. Thanks
1
vote
2answers
104 views

Prove $(A \backslash B)'\backslash (B \backslash A) = B \backslash A$

Let $A$ and $B$ be subsets of some universal set. Prove that $(A \backslash B)'\backslash (B \backslash A) = B \backslash A$ Given: Definition 3.3.1 states that $A$ and $B$ are sets. The complement ...
-1
votes
4answers
245 views

Prove $A \subseteq B \cap C$ if and only if $A \subseteq B$ and $A \subseteq C$

Prove the following for any sets $A,B$ and $C$. This is actually two sets that I'm trying the prove. The title character restriction wouldn't allow me to post both at the same time. a. $A \subseteq ...
1
vote
2answers
323 views

Verify a Tautology without a truth table.

Verify that the following are tautologies. Do not make truth tables. a. $\lnot(\lnot) P \leftrightarrow P$ The first question is just a double negation law. So, if I have to take the left side and ...
1
vote
3answers
71 views

Prove the following $f_{(A \cup B)}(x)=f_A(x)+f_B(x)-f_A(x)\cdot f_B(x)$

There is option to prove the following with truth table? $$f_{(A \cup B)}(x)=f_A(x)+f_B(x)-f_A(x)\cdot f_B(x)$$ I would like to get some hints how to do it in formal way(not truth table) thanks!
0
votes
0answers
55 views

Explain why the description defines a Boolean algebra.

Here is the exercise I am trying to figure out. Let A = {a,b} and list the four elements of the power set P(A). We consider the operations + to be $\cup$, . to be $\cap$, and complement to be set ...
0
votes
4answers
76 views

Simplifying this boolean function

How can I completely simplify this equation using algebraic simplification rules? $$x'y'z + x'yz + xyz$$
1
vote
2answers
59 views

How to simplify this using boolean algebra?

My paper is due tomorrow and there is only the last exercise left for me to do. However, I don't have any sufficient notes or examples on how to simplify it. Any help would be appreciated! A'B'C' + ...
1
vote
3answers
58 views

Where did I go wrong with this Boolean simplification?

I am completely new to Boolean algebra, and I've tried to simplify this expression. All I did is tried to follow my lecturers methods, but I don't think it's right, and I have no idea how to do it. ...
0
votes
0answers
140 views

Convert expression to NAND only

Endless youtube videos and reading through notes later I am yet again stuck. I have to covert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot ...
1
vote
4answers
98 views

Boolean Algebra simplify minterms

I have this equation $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot \bar{C} + A \cdot B\cdot C$$ and need to simplify it. I have got as far as I can and spent a good 2 ...
0
votes
1answer
68 views

simplify boolean algebra expressions

Use Boolean algebra to simplify the expression for F1, where, F1 = A’.B’.C’.D’ + A’.B’.C.D’ + A’.B’.C.D + A’.B.C’.D + A’.B.C.D’ + F2 and F2 = A’.B.C.D + A.B’.C’.D’ + ...
0
votes
1answer
85 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
125 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 $zw(x'+x)(y'+y)+z'w'(x'+x)(y'+y)+xy'z'w+xyz'w=(zwx'+zwx)(y'+y)+(z'w'x'+z'w'x)(y+y')+xy'z'w+xyz'w$ ...
1
vote
1answer
742 views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
1
vote
1answer
339 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
324 views

Disjunctive normal form expansion

I do not understand this at all. Find the sum-of-products expansions of these Boolean functions. $F(x, y, z) = x + y + z$ $F(x, y, z) = (x + z)y$ $F(x, y, z) = x$ $F(x, y, z) = x y$ ...
0
votes
2answers
123 views

Boolean algebra simplification

I have the equation (CD)+(BC'D'+A'B'C'D)+(C'D') and I can't seem to simplify it enough for my homework. I've tried multiple ways of simplifying the equation and I keep ending with an extra variable D. ...
0
votes
3answers
67 views

Three Boolean Algebra Proofs - I just don't get it!

I'm having a very difficult time proving the following 3 expressions: $$\begin{align*} &x\cdot y\cdot z+x'\cdot z=y\cdot z+x'\cdot z\\ &x\cdot y+y\cdot z+x'\cdot z=x\cdot y+x'\cdot z\\ ...
0
votes
0answers
64 views

Boolean Algebra problem

Given two binary vectors $A=(a_1,..,a_n), B=(b_1,..,b_n) \in \varphi^n$, we define the Hamming's distance between $A$ and $B$, $\rho (A,B)$ as: $\rho(A,B)=\sum_{i=1}^{n}|a_i-b_i|$ And, for a vector ...
1
vote
1answer
110 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 ...
-1
votes
1answer
89 views

boolean expression help simplify

Question 1 $$\begin{align*}A'BC + AB'C + ABC + A'B'C' &=A'B C + B' C (A +A') + ABC\\ &=A'B + C + B'C + ABC \end{align*}$$ Question 2 $$\begin{align*}A'B + B'C + CB &=A'B + C(B +B')\\ ...
1
vote
2answers
36 views

Question about poset and boolean ordering, inf and sup

We have a poset $(X, \sqsubseteq)$, and we define operations $+$ and $\cdot$ by $x+y=inf(x, y)$ and $x\cdot y=sup(x, y)$ ($+$ can be seen as union in sets and $\cdot$ as intersection in sets). The ...
2
votes
1answer
147 views

Solving Boolean Expressions with Theorems

I'm having the hardest time wrapping my head around this stuff. This is a homework problem, one of many. I just need some help on what to do. ...
0
votes
1answer
59 views

Correctness of answers and question about sup en inf

In $A=\{2, 3, 6, 12, 36, 72, 108\}$ we define the relation $R$ by $aRb$ if $b=a$, or $b=2a$ or $b=3a$. Q1: Draw the graph of $R$ and list which properties $R$ has. A1: The properties are: ...
0
votes
2answers
69 views

How to prove boolean ordering question

Let $\sqsubseteq$ be the boolean ordering of $X$, so for every $x$ and $y$ applies $x \sqsubseteq y$ if $x \sqcap y = x$. Let $v, w, a, b \in X$ with $v \sqsubseteq a$ and $w \sqsubseteq b$. Show that ...
0
votes
1answer
46 views

How do I solve this Boolean Algebra Problem?

Let A be an arbitrary but fixed Boolean algebra with operator $@$ and $*$ and $'$ and the zero and unit element be denoted by $0$ and $1$ respectively. let $x,y,z \in A$ if $a,y \in A$ such that ...
2
votes
1answer
39 views

Need help for right direction simplifying boolean algebra formula

I have the following boolean algebra, where union is $+$ and intersection is $\cdot$ : $(x\cdot y)+((z+y)\cdot \bar{z})+y=y$ Is there a systematic way of doing this, or do you need to puzzle? My ...
0
votes
1answer
46 views

Help with getting the right direction on a boolean algebra question

Need some help getting in the right direction for answering the following question: Prove the following property and interpret this in $\mathcal P \left ({V} \right)$: if $x+ \bar y=$ 1, then ...
0
votes
1answer
88 views

How are boolean expressions converted to NOR expressions?

What kind of rules help to convert an expression into a 3 input NOR expression? Do all variables have to be of the form (a+b+c)' + (d+e+f)'? What happens if there is an expression that is just (a')' ...
0
votes
0answers
57 views

Modify boolean equation to get 3 input NOR equation using boolean algebra rules

I was taking a look at this link http://lizarum.com/assignments/boolean_algebra/chapter3.html to try and solve an equation I have. The original equation is: H = MC + MC' + CRD + M'CD' I simplified ...
0
votes
1answer
78 views

Multiply the number $(1001)_{2}$ by 3 digit number

I want to multiply the number $(9)_{10} \rightarrow (1001)_{2}$ by a 3 digit binary number. 1) How I can extract the boolean equations? 2) Make a circuit of it. so what I did is just see what ...
1
vote
1answer
314 views

Logic Circuits And Equations Issue - Multiply Binary Number By 3

I am trying to build a logic circuit that multiplies any 4 digit binary number by 3. I know that if I multiply\divide number by 2 it moves left\right the digits, but what I`m doing with multiply by ...
0
votes
1answer
83 views

Logic Circuit Question

1) Write the boolean expression after every GATE 2) Write the boolean expression of GATE 3 3) Try to simplify the boolean expression of GATE3 I need to know if what I did its right + your advice ...