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 […]

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 […]

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?

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 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 […]

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 […]

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!

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 […]

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 […]

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 $p\vdash q$ […]

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