How do Dictionary<TKey,TValue> and HashSet<T> work, and what equality rules do their keys require?

Dictionary and HashSet hashing and equality in C#

Explains how hash-based collections use hash codes and equality, then chooses Dictionary or HashSet from the value the program needs to store.

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Dictionary and HashSet are hash-based collections. A dictionary maps a unique key to a value, while a hash set stores unique values and answers membership questions. Both use a hash code to narrow the search and equality to confirm a match. Different values can share a hash code, so the collection still needs equality after finding the bucket.

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Hash buckets, equality, comparers, unique keys, set membership, mutable keys, collisions, and average versus worst-case behavior.

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Why this question matters
Hash-based collections are everyday tools whose correctness depends on an equality contract that is easy to violate silently.
Dictionary and HashSet are hash-based collections. A dictionary maps a unique key to a value, while a hash set stores unique values and answers membership questions. Both use a hash code to narrow the search and equality to confirm a match. Different values can share a hash code, so the collection still needs equality after finding the bucket.
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A value that compares equal must return the same hash code for as long as it is stored. I avoid mutating fields that participate in key equality because the object may then live in the wrong bucket and become impossible to find normally. I choose an explicit comparer when equality is domain-specific, such as case-insensitive product codes. Lookup, insertion, and removal are usually O(1) on average, but collisions and resizing still have costs.

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Overriding Equals without a consistent GetHashCode.
Mutating a key after insertion.
Using Dictionary when only uniqueness is needed.
Relying on default string comparison without deciding the semantics.

How do Dictionary<TKey,TValue> and HashSet<T> work, and what equality rules do their keys require?

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