MATHS Common pitfalls

• Confusing an object with its notation. $\mathbb{R}$ is the real-number set; the symbol is just the conventional name.
• Writing $A\in B$ when you mean $A\subseteq B$. An element belongs to a set; a subset is contained in a set. For example, $\mathbb{N}\subseteq \mathbb{Z}$, not $\mathbb{N}\in \mathbb{Z}$.
• Ignoring the Domain. The same formula can define different functions on different domains, and this changes invertibility, differentiability, and physical meaning.
• Treating the codomain and range as automatically identical. They are identical only for a surjective function; see Codomain and range.
• Cancelling or dividing by an expression that might be zero. Always state the non-zero condition.
• Forgetting units in physics applications. $x=3$ is incomplete if the quantity is a length; $x=3[[Metre|m]]$ carries different information from a dimensionless 3.
• Assuming a picture proves the claim. Diagrams are excellent intuition, but proofs need definitions or a controlled approximation.
• Treating approximate equality as exact equality. Numerical work, measurement, and series truncation need explicit error awareness.

Good default check: identify the set, the function or relation, the domain, and the units before manipulating symbols.