Articles of predicate logic

$a$ is not equal to $b$. How to express it in predicate logic notation

Suppose I have an Excel (or a csv) file with 2 columns: $a$ and $b$. For each row I would like to forbid the following: $a\neq b$. So for instance in row 1 I cannot have $a=3$ and $b=3$. However, the question is not how to implement it in Excel. The question is how to […]

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 […]

What is a Primitive Atomic Formula?

I am reading “Axiomatic Set Theory” by Patrick Suppes and he defines a primitive atomic formula as follows: A primitive atomic is an expression of the form ($v\in w$ ), or of the form ($v=w$) where v and w are either general variables or constant ‘0’(empty set). My question is what is a primitive atomic […]

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn’t be formal, but in the “usual” theorem-proving style in mathematics) Of course I’ve given it a try and came up with this: Proof: Suppose $\neg \exists x(P(x) \Rightarrow \forall […]

Prove for all sequences $\{a_n\}$ and $\{b_n\}$, if $\lim a_n = a$ and $\lim b_n = b$, then $\lim a_n + b_n = a+b$ entirely in first-order logic

I already know how to prove this statement in “English,” but I would to see a proof of it entirely in first-order logic. Here is the English proof: (1) Let $\{a_n\}$ and $\{b_n\}$ be arbitrary sequences. (2) Assume $\lim_{n\to\infty} a_n = a$ and $\lim_{n\to\infty} b_n = b$. (3) This means for every $\epsilon_a >0$ there […]

How to show the statement is false?

How to do you show the statement is false and prove its negation is true? $$\forall n \in\mathbb Z^+ \exists a \in\mathbb Z^+ \text{ such that } a|n\text{ and }\frac na\text{ is odd}$$

Analyzing the logical form of “All married couples fight”

This is one of the example problems in Velleman’s How to Prove book: Analyze the logical forms of the following statements. All married couples have fights. Solution: ∀x∀y(M(x, y) → F(x, y)), where M(x, y) means “x and y are married to each other” and F(x, y) means “x and y fight with each other.” […]

Pre-nex normal form. Correct way to distribute negations among quantifiers

Start point: $$(¬∀x P(x) ∨ ¬∀y Q(y)) → ¬∃x G(x)$$ Implication to Disjunction (DeMorgans Laws): $$¬(¬∀x P(x) ∨ ¬∀y Q(y)) ∨ ¬∃x G(x)$$ Now I am at the point where I need to move in the negations to precede the quantifiers but there are two ways I can see that i can do this to […]

Negating an existential quantifier over a logical conjunction?

I was wondering how one would negate an existential quantifier over a logical conjunction. As an example, suppose that I have the statement: “There exists a car that is white and doesn’t use diesel”. I am recognizing this as saying: $\exists x \in C: \ (P(x) \land Q(x))$, where $C$ is the set of all […]

Predicate logic proof problem

Where the domain of the variables are Real Numbers, determine the truth value for the following: $$ \forall x \exists y(y^2-x<200) $$ I don’t understand how to formally prove this problem. Since $y^2\geq 0$ it would stand to reason that any $x< 0$ could disprove this statement. I tried to prove it true by using […]