(Logic and proof)=

# Logic and proof#

Mathematics for Geoscientists

## Logic symbols#

The origins of logic symbols stem from first attempts at creating a clear language to express logic statements. Everyday languages are not suitable for this as the use of them can easily lead to ambiguities and paradoxes. This is the case in mathematics as well, so general mathematics was influenced by this language and borrowed some of it. The use of logic symbols is more widespread in areas of mathematics where there is a need for a clear language, for example set theory (which is full of paradoxes!).

Here we will list some more common logic symbols. For a full list visit Wikipedia.

Table 1 Common logic symbols#

Symbol

Name

Read as

Example

Meaning

$$\forall$$

universal quantification

for all, for any, for each

$$\forall x \in \mathbb{R}, x^2 \geq 0$$

For all real numbers $$x$$, $$x^2$$ is greater or equal to $$0$$

$$\exists$$

existential quantification

there exists

$$\forall n \in \mathbb{N}, \exists x \in \mathbb{R} : | n - x | < 1$$

For all natural numbers $$n$$ there exists a real number $$x$$ such that $$| n - x| < 1$$

$$\exists !$$

uniqueness quantification

there exists exactly one

$$A \subseteq \mathbb{N} \implies \exists ! m \in A : m \geq n, \forall n \in A$$

If $$A$$ is a subset of $$\mathbb{N}$$ then there exists a unique $$m \in A$$ such that $$m \geq n$$, for all $$n \in A$$

$$\lnot$$

negation

not

$$\lnot (x = y) \iff (x \neq y)$$

$$x = y$$ is not true iff $$x \neq y$$

$$\land$$

conjunction

and

$$((x \leq 1) \land (x \geq 1)) \iff x = 1$$

$$x$$ is less or equal than 1 and greater or equal to $$1$$ iff $$x = 1$$

$$\lor$$

disjunction

or

$$((x \leq 0) \lor (x \geq 0)) \iff x \in \mathbb{R}$$

$$x \leq 0$$ or $$x \geq 0$$ iff $$x$$ is a real number

$$\implies$$

implication

implies, if… then

$$n \in \mathbb{N} \implies n \in \mathbb{Z}$$

If $$n$$ is element of $$\mathbb{N}$$ then it is element of $$\mathbb{Z}$$

$$\iff$$

equivalence

if and only if, iff

($$x$$ is prime) $$\land (x\mod 2 = 0) \iff x=2$$

$$x=2$$ if and only if $$x$$ is a prime number and $$x \mod 2 = 0$$

$$:$$, $$|$$, s.t.

condition

such that

$$\forall x \in \mathbb{R} : x > 0, \sqrt{x} > 0$$

For all $$x$$ element of $$\mathbb{R}$$ such that x is greater than $$0$$, $$\sqrt{x}$$ is greater than $$0$$