AI Tutor
Imagine writing x=3  ⟹  x2=9  ⟹  x=3x = 3 \implies x^2 = 9 \implies x = 3. That looks harmless at first glance, but you just claimed −3=3-3 = 3.
In higher mathematics, symbols are not decorative bullet points — they are exact logical statements.

Implication vs. Equivalence

The implication symbol (P  ⟹  QP \implies Q) means 'if statement PP is true, then statement QQ must be true' — the logic flows only left to right.
The equivalence symbol (P  ⟺  QP \iff Q), read as if and only if (or iff), means the logical flow works in both directions (P  ⟹  QP \implies Q and Q  ⟹  PQ \implies P).
What happens when our statements depend on which numbers we are allowed to choose?

Quantifiers and Domain Restrictions

The symbol ∀\forall stands for for all (universal quantifier), while ∃\exists stands for there exists (existential quantifier).
The order of quantifiers changes the entire meaning: ∀x∈R,∃y∈R such that y>x\forall x \in \mathbb{R}, \exists y \in \mathbb{R} \text{ such that } y > x is true, but ∃y∈R,∀x∈R such that y>x\exists y \in \mathbb{R}, \forall x \in \mathbb{R} \text{ such that } y > x claims a single number is bigger than every number.
How do we weave these symbols together into a readable written argument?

Structural Connecting Language