-1
votes
2answers
20 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
17 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
34 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, ...
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 ...
0
votes
1answer
49 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 = ...
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 ...
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
0answers
27 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 ...
1
vote
1answer
48 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
vote
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 : ...
1
vote
0answers
34 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
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 ...
0
votes
2answers
24 views

Boolean Algebra Simplification

How do I simplify the following equation? $\newcommand{\pn}{\phantom{\neg}}$ $$\begin{align*} \neg A\pn B \neg C \neg D\\ + \pn A\neg B\neg C\neg D\\ + \neg A\neg B\neg C\pn D\\ + \pn A\pn B\neg C\pn ...
1
vote
3answers
48 views

Show that (P→Q) ∧ (Q→R) is equivalent to (P→R) ∧ [(P↔Q) ∨ (R↔Q)]

I literally have no idea how to start this proof. I get to (P→Q) ∧ (Q→R) = (¬P ∨ Q) ∧ (¬Q ∨ R) and then I get stuck.
0
votes
1answer
33 views

Simplifying a logic function using boolean algebra

I have the the following logic function (where $'$ is NOT) $f(a, b, c) = abc + ab'c + a'bc + a'b'c + ab'c'$ I have to simplify it as much as possible using only boolean algebra (no truth tables, ...
1
vote
3answers
63 views

The negation of an implication statement

$$\neg(A \longmapsto B)\lor \neg B$$ Does this this expression simplify to:? $$\neg A\longmapsto\neg B\lor \neg B$$ Which further simplifies to: $$\neg A\longrightarrow\neg B$$
0
votes
1answer
61 views

simple exercise in Cylindric algebra

I am trying to gain a better understanding of cylindric algebra, so I made up this example. Given a general rule that someone's father's father is his/her grandfather: $\forall_X ~ \forall_Y ~ ...
0
votes
0answers
37 views

XOR with multiply operation.

can I do that $((A*5) \oplus A)==A*(5\oplus1)?$ and that $(A \oplus B/2) == ((2*A) \oplus B)$? Thanks.
2
votes
2answers
51 views

'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$

For me, it is pretty clear that $a + \overline{a}*b = a + b$, because the first $a$ in the or will make sure that if the second term must be 'evaluated', $a$ will ...
0
votes
2answers
77 views

Finding boolean/logical expressions for truth table + explanation [closed]

I'm having very hard time finding boolean expressions from truth tables. I've also tried many tricks but still can't get through...think you guys can help me with this??...you'll be doing this lil ...
1
vote
1answer
53 views

Where do I start with $\sim((P\wedge Q)\vee \sim(P\vee Q))$?

can anyone tell me in a table form how to start with this $\sim((P\wedge Q)\vee \sim(P\vee Q))$ I am confused on how to do this part $\sim(P\wedge Q)$, which one we do first, inside brackets or ...
-1
votes
1answer
49 views

how to apply laws of boolean algebra to solve boolean expression [closed]

V=(A+B+C) . (A'+B'+C'). A How to simplify above Boolean-Expression,How to apply Boolean laws
0
votes
3answers
97 views

Simplifying a Sum of Products expression

I'm having some trouble with reducing the Sum of Products expressions for some questions on an upcoming exam. Below is the table (which is correct) for the first part of the question, the second part ...
1
vote
1answer
38 views

The ability of a logical statement to represent a two-place truth function.

How can i determine which two-place truth functions can be represented using a logical statement built out of a subset of two logical connectors in $ \{\rightarrow, \wedge, \vee ,\equiv \}$ ? for ...
0
votes
1answer
46 views

Converting large terms to disjunctive normal form (logic)

So hello everyone, I am doing some boolean logic and I have this exercise to convert the following term to DNF (disjunctive normal form), but it is so large that everything I try ends up being mega ...
5
votes
1answer
73 views

Is it possible to express “$P\leftrightarrow Q$” as a formula in $\to,\neg$ with $P$ only appearing once?

I want to write a propositional logic formula for the biconditional that only uses one side of the biconditional once in the formula. I expect it is impossible, but can anyone think of a proof? There ...
0
votes
3answers
26 views

How to define equivalency as NAND only function?

I am struggling a bit with boolean algebra. I need to represent equivalency as NAND only function. $(A * B) + (-A * -B)$ I am trying with the Morgan rule but I don't know if I can do that: $(A * ...
1
vote
2answers
55 views

Truth-functional completeness

Let the statement $?PQR$ be determined by the following truth-table. ...
3
votes
2answers
70 views

Boolean Logic - Reduction - $a \vee (a \wedge b) = a$

How would I simplify / reduce the following equation using boolean identities/proofs? $$a \vee (a \wedge b) = a$$ So far I've used the distributivity identity and got $$(a\vee a) \wedge (a\vee b)$$ I ...
2
votes
3answers
58 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, ...
1
vote
1answer
31 views

How can i turn the Boolean Equation pq+r into a switch circuit?

How can I turn the Boolean Equation $pq+r$ into a switch circuit? I have synthesized this and drawn the NOR gates circuit however I'm not sure how to go about drawing/constructing the switch circuit.
0
votes
1answer
26 views

Applying De Morgans Laws to $a+bc+\overline{a}b\overline{c}d$ in terms of the NOR operator

I need to synthesize $f=a+bc+\overline{a}b\overline{c}d$ into the NOR form. Can I split this since I know that $a+bc=(a+b)(a+c)=\overline{\overline{a+b}+\overline{a+c}}$? I'm just not sure how to go ...
2
votes
1answer
44 views

Applying De Morgan's to express $pq+r$ in terms of NOR operator

In Boolean Algebras I have $pq+r$ which I think is the same as $(p+r)(q+r)$. Now, I need to use De Morgan's laws to synthesize this into the NOR form but I am not sure how to apply the laws here.
2
votes
2answers
273 views

Proving hypothetical sylloligism (p implies q, q implies r, therefore p implies r) with boolean algebra

I'm trying to prove the hypothetical sylloligism using boolean algebra. We already have a solution using propositional logic, which relies on proof by contradiction. $(p \implies q) \wedge (q ...
1
vote
1answer
55 views

Semantic consequence

I'm studying refutation trees in computer science II, but I have a big doubt: Let $\Gamma, \Psi \subseteq F_m$ Is the following hypothesis true? $\Gamma \vDash \Psi \iff \not\vDash\{\Gamma,\lnot ...
1
vote
2answers
47 views

Boolean Expression simplification help

Hi I am new to the board. Taking a computer architecture course and I am having trouble understanding further simplification on a question I got on a previous quiz. When I type in the expression ...
1
vote
1answer
30 views

Can covering be done on two elements?

The covering rule is: $$B \bullet (B+C) = B$$ and $$B+(B \bullet C)=B$$ So does it follow from this rule that: $$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet ...
2
votes
0answers
68 views

Simplify Product of Sums

Similar question to: Boolean Algebra - Product of Sums I was given a truth table and asked to give the sums-of-products and the product-of-sums expressions. I reduced the sums-of-products ...
2
votes
3answers
75 views

Construct XNOR with only OR gates

Is it possible to construct the XNOR gate which is given as, a XNOR b = (a AND b) OR (~a AND ~b), by using only OR gates. So from the definition, the question boils down to: can you construct the AND ...
1
vote
1answer
57 views

Alternative to xor(A,B,C)

How can we make a comprehensive statement, which will correspond to the truth table of xor (A, B, C) by combining logical operators AND (&), OR (|), XOR (xor) and NOT (!)?
2
votes
1answer
90 views

Convert a boolean function into K-map

I would like to know how can I convert the following boolean function into a truth table and accordingly construct the k-map $$F = A'B'C'+B'CD'+A'BCD'+AB'C'$$ thanks in advance :)
0
votes
0answers
81 views

Boolean algebra - cube - minimal disjunctive normal form

I have a test coming up and I would like to know how to solve these kinds of problems. This is the description: ...
4
votes
3answers
146 views

Peculiar examples to the Stone Representation Theorem

The Stone Representation theorem states that every Boolean algebra is isomorphic to a field of sets. That is, a Boolean algebra whose elements are sets, and sums, products, negation are union, ...
1
vote
5answers
69 views

If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
2
votes
3answers
60 views

Proving $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$.

How do I prove using boolean algebra that $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$? I can see it in the logic table and it is logical, but I can't prove it mathematically.
0
votes
1answer
60 views

How can I simplify this boolean equation for the multiplexer a little further?

I've obtained a formula through cannonical representation, which is: $$A\cdot \overline{B\cdot S}+A\cdot B\cdot \overline{S}+\overline{A}\cdot B\cdot S+A\cdot B \cdot S$$ And I'm trying to simplify ...
0
votes
1answer
69 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) ...
1
vote
1answer
117 views

Conjunctive Normal Form vs Product of Sums

I am confused as to what the difference between Conjunctive Normal Form and Product of Sums is. Can someone explain what is different about them? It seems like they both only use groups of OR ...
0
votes
2answers
92 views

How to construct the truth table for a combinational circuit

I am trying to construct the truth table for a combinational circuit with the following conditions : A) Room with 4 doors , 1 light, a switch near each door that controls the light (4 in total) B) ...