Intereting Posts

If D is an Integral Domain and has finite characteristic p, prove p is prime.
Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues
Inverse of a composite function from $\mathbb R$ to $\mathbb{R}^p$ to $\mathbb R$ again with a non-zero continuous gradient in a point
$GL(n,K)$ is open in $M(n,K)$
The logic behind the rule of three on this calculation
Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$
There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$
An extension of a game with two dice
Asymptotic behavior of $\sum\limits_{n=1}^{\infty} \frac{nx}{(n^2+x)^2}$ when $x\to\infty$
Laplace transform of $f(t^2)$
Embeddings are precisely proper injective immersions.
Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction
Logic question: Ant walking a cube
Prove that the following set is dense
Local and global logarithms

I’ve done a truth table after reducing it to this and it seems to be equal to $\neg Q$:

$$\lnot Q \lor (\lnot Q \land R) = \lnot Q$$

But when I try to show it without a truth table (with just transformations), I end up in a loop between that and:

- Defining Category of Problems
- Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
- Negation of quantifiers
- $\omega$-stable theories in finite relational language
- Theorems that we can prove only by contradiction
- 'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$

$$\lnot Q \land (\lnot Q \lor R)$$

Is there a way to show this is true without using a truth table? What am I missing?!

Thanks in advance!

- Proving ${\sim p}\mid{\sim q}$ implies ${\sim}(p \mathbin\& q)$ using Fitch
- Some questions about mathematical content of Gödel's Completeness Theorem
- What does $\rightarrow$ mean in $p \rightarrow q$
- Proof of a theorem in Hilbert's system
- Proof by contradiction and Incompleteness
- Can every proof by contradiction also be shown without contradiction?
- What precisely is a vacuous truth?
- For any set of formulas in propositional logic, there is an equivalent and independent set
- Is “A and B imply C” equivalent to “For all A such that B, C”?
- Does Chaitin's constant have infinitely many prime prefixes?

\begin{align*}

\neg Q \vee (\neg Q\wedge R)

&= (\neg Q\wedge 1) \vee (\neg Q\wedge R) \\

&= \neg Q \wedge (1\vee R) \\

&= \neg Q \wedge 1 \\

&= \neg Q

\end{align*}

When it comes to showing that logical statements are independent of certain variables I like to cycle through them and show it this way. Notice that you’ve shown that your statement is independent of the truth or falsity of $R$ so I’ll take that approach here.

Let’s suppose $R = T$, then

$$\neg Q\vee(\neg Q\wedge R) = \neg Q\vee(\neg Q\wedge T) = \neg Q\vee (\neg Q)= \neg Q.$$

Likewise if $R = F$, then

$$\neg Q\vee(\neg Q\wedge R) = \neg Q\vee(\neg Q\wedge F) = \neg Q\vee F = \neg Q.$$

Thus no matter what the truth value of $R$ is we get $\neg Q$ so we are forced to conclude that $\neg Q\vee(\neg Q\wedge R) = \neg Q.$

$Q\implies\lnot Q \wedge R\implies\lnot Q$

Contradiction!

Assume that Q is true. Then it follows that $\lnot$Q is false, and ($\lnot$Q$\land$R) is false also. So, ¬Q∨(¬Q∧R) is false.

Assume that Q is false. Then $\lnot$Q is true, and thus so is ¬Q∨(¬Q∧R).

Since Q is either true or false, and since in either case the equality here ¬Q∨(¬Q∧R)=¬Q, it follows that ¬Q∨(¬Q∧R)=¬Q in all cases.

If you don’t mind using order theory, you can show it without any reference to true or false. Let $\leq$ be an order with $A \vee B \geq A$ and $A \wedge B \leq A$. Then, $\neg Q \vee (\neg Q \wedge R) \geq \neg Q$ and $\neg Q \wedge (\neg Q \wedge R) \leq \neg Q$. If the two expressions are equal, you can see that they must equal $\neg Q$.

$

\newcommand{\calc}{\begin{align} \quad &}

\newcommand{\calcop}[2]{\\ #1 \quad & \quad \unicode{x201c}\text{#2}\unicode{x201d} \\ \quad & }

\newcommand{\endcalc}{\end{align}}

\newcommand{\ref}[1]{\text{(#1)}}

\newcommand{\true}{\text{true}}

\newcommand{\false}{\text{false}}

$Here is yet another way to calculate this:

$$\calc

\lnot Q \lor (\lnot Q \land R) \;\equiv\; \lnot Q

\calcop\equiv{both $\;X \lor Y \equiv Y\;$ and $\;\lnot X \lor Y\;$ are alternative ways to write $\;X \Rightarrow Y\;$}

\lnot (\lnot Q \land R) \lor \lnot Q

\calcop\equiv{DeMorgan}

Q \lor \lnot R \lor \lnot Q

\calcop\equiv{excluded middle; simplify}

\true

\endcalc$$

- A problem about convergence…
- Are there open problems in Linear Algebra?
- Irreducible and prime elements
- Composition of linear operators and ranks…
- Show that $\liminf_{n\to \infty}x_{n}\le\alpha(x)\le\limsup_{n\to\infty}x_{n}$ for $x=(x_{n})$ in $\ell^{\infty}$
- Polynomial maximization: If $x^4+ax^3+3x^2+bx+1 \ge 0$, find the maximum value of $a^2+b^2$
- Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$
- Splitting field of $x^6+x^3+1$ over $\mathbb{Q}$
- Show that $\Bbb Q/\Bbb Z $ is an infinite group?
- Prove that there exists bipartite graph with this degree sequence: $(3,3,3,3,3,5,6,6,6,6,6,6,6,6)$
- Is the collection of finite subsets of $\mathbb{Z}$ countable?
- Commuting matrix exponentials: necessary condition.
- How can I find $\lim_{n\to \infty} a_n$
- Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$
- how to find integer solutions for $axy +bx + cy =d$?