Set theory Common pitfalls

• Using $\in$ for subsets. Write $\mathbb{N}\subseteq \mathbb{Z}$, not $\mathbb{N}\in \mathbb{Z}$. Use $\in$ when the left-hand side is an element.
• Forgetting that arnothing is a subset of every set, including itself.
• Confusing $\{x\}$ with $x$. A box containing an object is not the same thing as the object.
• Assuming $A\setminus B=B\setminus A$. Set difference is not symmetric.
• Treating a Relation as a Function without checking the “exactly one output for each input” condition.
• Saying a function is “onto” without naming the codomain. Surjectivity depends on the chosen target set.
• Forgetting that intervals have endpoint conventions: $(a,b)$ excludes endpoints, $[a,b]$ includes them.
• Assuming all infinities behave like finite sizes. Infinite sets need cardinality arguments; intuition from finite counting can fail.
• In probability or physics, forgetting to specify the sample space or state space. Events and states only make sense relative to an ambient set.

Reliable check: write the ambient set first, then ask whether each claim is about membership, subset inclusion, or a mapping.

A useful debugging move is to translate a sentence into symbols and then back into words. If the translation changes meaning, the issue is usually membership versus subset, relation versus function, or codomain versus range.