Articles of propositional calculus

Finish proof of tautology

I’m new to Discrete math, I’ve been working on this problem for near an hour and can’t figure out the next steps. The problem is: Show that $[\neg p \land(p \lor q)] \to q$ is a tautology. Here is what I have so far. \begin{align} \neg[\neg p \land(p \lor q)] \lor q &≡ [p \lor […]

Every $n$-ary logical connective has a DNF

I’m trying to solve the following exercise: Let $A_1,…,A_n$ be propositions, $n \in \mathbb N$. Show that every $n$-ary logical connective $J(A_1,…,A_n)$ considered as a function $J:\{t,f\}^n \to \{t,f\}$ ($t$ is true, $f$ is false) has a disjunctive normal form, in other words can be represented as a disjunction $Y_1,…Y_k$ of a suitable amount of […]

Proving ${\sim p}\mid{\sim q}$ implies ${\sim}(p \mathbin\& q)$ using Fitch

I am struggling with proving something in Fitch. How can I prove from the premise ~p | ~q , that ~(p & q). Any ideas on how I should proceed?

Is this conclusion via rules of inference correct?

Use rules of inference to show: ∀x(P(x) → Q(x)) premise ∀x(Q(x) → R(x)) premise ¬R(a) premise ¬P(a) conclusion I have a lot of trouble with these sort of questions and was wondering if I did this correctly. Usually I have no idea which rules to use and it feels like I just need to try […]

How can the completeness of Hilbert's axioms be proven?

How can one prove that every propositional tautology, expressed with the connectives ‘$\neg$’ and ‘$\rightarrow$’, can be proved with the axioms below? (P0. $\phi \to \phi$) P1. $\phi \to \left( \psi \to \phi \right)$ P2. $\left( \phi \to \left( \psi \rightarrow \xi \right) \right) \to \left( \left( \phi \to \psi \right) \to \left( \phi \to […]

Proving De Morgan's Law with Natural Deduction

Here is my attempt, but I’m really not sure if I’ve done it right; as I’m just about getting the hang of Natural Deduction technique. Have I done it correctly? If not, where did I make errors and how should I do it? Thank you in advance! Sorry for the bad image quality; I’m bad […]

Why is $p\Rightarrow q$ equivalent to $\neg p\lor q$ and how to prove it

I don’t know how to prove that $p\Rightarrow q$ is equivalent to $\neg p\lor q$ ,here is the link p=>q . And I don’t know how wolframalpha generate “Minimal forms” . Can you prove $p\Rightarrow q \equiv \neg p\lor q$, and explain how to get “Minimal forms” ? Thanks!

How to show $\vdash (\neg\neg p \rightarrow p)$.

Given these axioms: where $\phi, \psi, \theta$ are formulas $$ 1.:(\psi \rightarrow (\theta \rightarrow \psi))$$ $$ 2.: ((\neg \psi \rightarrow \neg \theta) \rightarrow (\theta \rightarrow \psi))$$ And using the deduction theorem. So I started with trying to show that $\neg\neg p \vdash p$ (so I can use deduction theorem later). From this: 1.$\neg\neg p$ Assumption […]

Finding a logical expression (under some constrains) s.t. it is equivalent to another one

In this question, it was made clear, when $\bullet$ some statement $A$ is stronger than another statement $B$, namely if $A\Rightarrow B$ holds; and when the statement $A$ is weaker than another statement $B$, namely if $B\Rightarrow A$ holds. and $\bullet$ when a theorem $A\Rightarrow B$ (every mathematical theorem is from the point of view […]

formal proof – logic

I am trying to prove the following, using natural deduction: $$p\wedge q\Leftrightarrow p \vdash p \Rightarrow q$$ with the following but i seem to get stuck. I know i have to prove $q$, but am not sure if this does it. can anyone help me please? thank you. $p\wedge q\Leftrightarrow p$ assumption, 0 $p\vdash q$ […]