**Introduction:**

Set Theory involves sets which are unordered lists.

Ex. Set: **S **= (a, b, c)

Note: **S** is the set and a, b, c are the elements of the set.

- All members of a set must be of the same type. (Note: because set can have complex types (sets themselves); this can be circumvented by the properties of the complex types.)

**Set Membership**:

**∅ element of****∅ = {}****∅**denotes the empty set

**∈ element of**- Set:
**S**= {a, b, c} - a ∈
**S**

- Set:
**∉ not element of**- Set:
**S**= {a, b, c} - d ∉
**S**

- Set:
**⊂ proper (strict) subset of**- Set:
**S**= {a, b, c} - Set:
**T**= {a, b} **T**⊂**S**

- Set:
**⊄ not proper (strict) subset of**- Set:
**S**= {a, b, c} - Set:
**Q**= {a, b, c} - Set:
**W**= {b, c, d} **Q**⊄**S****W**⊄**S**

- Set:
**⊆ subset of**- Set:
**S**= {a, b, c} - Set:
**Q**= {a, b, c} - Set:
**T**= {a, b} **Q**⊆**S****T**⊆**S**

- Set:
**⊈ not subset of**- Set:
**S**= {a, b, c} - Set:
**W**= {b, c, d} **W**⊄**S**

- Set:
**⊃ 𝐩𝐫𝐨𝐩𝐞𝐫 (𝐬𝐭𝐫𝐢𝐜𝐭) 𝐬𝐮𝐩𝐞𝐫𝐬𝐞𝐭 𝐨𝐟**- Set: S = {a, b, c}
- Set: T = {a, b}
- S ⊃ T

**⊅ 𝐧𝐨𝐭 𝐩𝐫𝐨𝐩𝐞𝐫 (𝐬𝐭𝐫𝐢𝐜𝐭) 𝐬𝐮𝐩𝐞𝐫𝐬𝐞𝐭 𝐨𝐟**- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: W = {b, c, d}
- Q ⊅ S
- W ⊅ S

**⊇ 𝐬𝐮𝐩𝐞𝐫𝐬𝐞𝐭 𝐨𝐟**- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: T = {a, b}
- Q⊇S
- T⊇S

**⊉ 𝐧𝐨𝐭 𝐬𝐮𝐩𝐞𝐫𝐬𝐞𝐭 𝐨𝐟**- Set: S = {a, b, c}
- Set: W = {b, c, d}
- W⊉S

**Set Operators:**

**∩ 𝐢𝐧𝐭𝐞𝐫𝐬𝐞𝐜𝐭𝐢𝐨𝐧𝐬**- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S ∩ T = (c, d, e, f)

**∪ 𝐮𝐧𝐢𝐨𝐧**- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S ∪ T = (a, b)

**\ difference (relative complement)**- S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S \ T = (c, d)
- Note: (c, d) is the relative complement
- T \ S = (e, f)
- Note: (c, d) is the relative complement

**a𝐛𝐬𝐨𝐥𝐮𝐭𝐞 𝐜𝐨𝐦𝐩𝐥𝐞𝐦𝐞𝐧𝐭**- Set: S = (a, b, c, d, e, f)
- Set: T = (a, b)
- Set: W = (c, d)
- T ⊂ S
- T ⊂ W
- The absolute complement is S \ (𝐓 ∪ W) = (e, f)

**∆ 𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐢𝐜 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧ce**- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f) “(S \ ” 𝐓)∪ (T \ S) = (c, d, e, f

**X Cartesian product**- Set: S = (a, b, c)
- Set: T = (d, e, f)
- S X T = ((a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f))
- T X S = ((d, a), (e, a), (f, a), (d, b), (b, e), (b, f), (c, d), (c, e), (c, f))
- Note: The Cartesian product operator produces a set of all possible ordered pairs between two sets.

**𝐏(𝐒) Power set**- Set: S = (a, b)
- P(S) = ((a, b), (a), (b), ∅)
- Note: The Power set of S produces the set of all subsets sets including the empty subset.
- between two sets.

**Cardinality**- Set: S = (a, b)
- Cardinality of set S is 2
- Note: The carnality of a set is the number of members in the set.

- 𝐈𝐧𝐬𝐞𝐫𝐭
- Set S = (a, b, c)
- Insert(S, d) returns S = (a, b, c, d)
- The Insert operator (i.e. Insert(Set, element)) adds a member to a set.

- 𝐃𝐞𝐥𝐞𝐭𝐞
- Set S = (a, b, c)
- Delete(S, a) returns S = (b, c)
- The Delete operator (i.e. Delete(Set, element)) removes a member from a set.

- 𝐋𝐨𝐨𝐤𝐮𝐩
- Set S = (a, b, c)
- Lookup(S, c) returns TRUE
- Lookup(S, d) returns FALSE
- The Lookup operator (i.e. Lookup(Set, element)) determines if an set contains a specific member.

**Common Sets: **

**ℕ**- The set of all natural numbers (0, 1, 2, 3, 4, 5….)

**ℤ**- The set of all integers (….-3, -2, -1, 0, 1, 2, 3…)

**ℚ**- The set of all rational numbers

**ℝ**- The set of all real numbers