# Tag Info

### Relations that ensure continuity

This answer was simplified by a lot from the original one. Consider a relation defined as follows: $x \sim y$ if $x > y$, or $y = x + 2^{m}$ for some $m\in\mathbb{Z}$. The idea is that the first ...
Accepted

Accepted

### Confused about saying every "statement" is vacuously true in an inconsistent arithmetic (terminology)

I think this is just s function of how we talk about theorems in mathematics in general. That is, whenever we prove something in mathematics we regard that theorem to be ‘true’ … though you are right ...

1 vote

### Understanding implication

The statement if A, then B is always true when A is false. Suppose that A is false. Then as ...
1 vote

### Finding the relationship (equivalence or implication) between two expressions

To check the conditional relationship between two sentences, making sense of them is usually faster than formal proof methods: $$\exists X \; (p(X) ∧ q(X))$$ Some object is both pink and quirky. \...
1 vote
Accepted

### Finding the relationship (equivalence or implication) between two expressions

The formula $\exists x Px \wedge \forall x Qx$ states "There exists at least one $x$ such that $x$ is $P$, and for every $x$, $x$ is $Q$." The formula $\exists x [Px \wedge Qx]$ states "...
1 vote

### Finding the relationship (equivalence or implication) between two expressions

The second implies the first. Suppose we have the second already. Then we can get in our context the following $x_0 \in X$ and $p(x_0)$ as well as $\forall x \; q(x)$. Combining the first and third ...
1 vote
Accepted

### Tautological statements

Since no one seems to be mentioning it and since I'm actually having some trouble finding an English-language source for what I'm talking about, I'll describe here the method of reductio ad absurdum. ...
1 vote

### How to show that $p∨q, q∨r, p\to ¬r ⊢ q$?

For the sake of variety, I offer a proof by contradiction. \$ \begin{array}{lllll} \{ 1 \} & 1. & p ∨ q & \text{premise} & \\ \{ 2 \} & 2. & q ∨ r & \text{premise} & \\ \...

Only top scored, non community-wiki answers of a minimum length are eligible