Membership versus inclusion: ∈ and ⊆
∈ means “is an element of.” For A = {1, 2, 3}, 2 ∈ A and 4 ∉ A. ⊆ means “is a subset of”: {1, 2} ⊆ A. Membership relates an element to a set; inclusion relates two sets. The element 2 and the singleton set {2} are different objects.
Union, intersection and difference
For A = {1, 2} and B = {2, 3}, A ∪ B = {1, 2, 3}, A ∩ B = {2}, and A ∖ B = {1}. Union collects elements in either set; intersection keeps shared elements; difference keeps those in A but not B. Set difference is generally not commutative.
Empty sets and proper subsets
∅ has no elements, while {∅} has one element. A ⊆ A is always true. Some authors use ⊂ for inclusion, others for proper inclusion. Use ⊊ when you need to exclude equality explicitly and state your convention.
LaTeX commands
\in → ∈; \notin → ∉; \subseteq → ⊆; \cup → ∪; \cap → ∩; \emptyset → ∅. Use these commands in math mode.
Check your understanding
For A = {a, b}, all three statements a ∈ A, {a} ⊆ A, and ∅ ⊆ A are true. They express membership, inclusion, and the fact that the empty set is a subset of every set.
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