Complement

Definition

The complement of a set contains all elements in the domain that are not in the set.

Example:

Domain = {1, 2, 3, 4, 5}
A = {2, 4}

Complement of A:

{1, 3, 5}

Complement = everything in the domain outside the set.

---

Domain Matters

Complement depends on the domain.

Example:

A = {2, 4}

If:

Domain = {1, 2, 3, 4, 5}

then:

Complement of A = {1, 3, 5}

If:

Domain = {1, 2, 3, 4, 5, 6}

then:

Complement of A = {1, 3, 5, 6}

Complement is only clear after the domain is clear.

---

Complement vs Difference

Complement is a special kind of difference.

Example:

Domain = U
Set = A

Complement of A means:

U minus A

It removes A from the whole domain.

Complement is difference from the full domain.

---

Complement And NOT

Complement is similar to logical NOT.

Membership in the complement means:

NOT in A

Example:

x is in complement of A

means:

x is in the domain AND x is not in A

Complement is set-level negation relative to a domain.

---

Final Summary

• The complement contains everything in the domain not in the set.
• Complement depends on the domain.
• Complement is difference from the full domain.
• Complement behaves like NOT for membership.
• Complement makes exclusion precise.