The document defines basic concepts in logic and set theory, including:
- What a set is and examples like sets of numbers
- Set operations like union, intersection, subsets, and complements
- Properties of sets like idempotence, commutativity, associativity, and distributivity
- Relations as subsets of Cartesian products, with examples of binary relations
9. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
10. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
11. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø empty set
12. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø
U
empty set
universe
13. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø
U
empty set
universe
Membership
a is a member of set A
14. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
10 ∈ {10, 23, 32}
Ø
U
empty set
universe
Membership
a is a member of set A
15. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
10 ∈ {10, 23, 32}
-1 ∉ N
Ø
U
empty set
universe
Membership
a is a member of set A
18. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Every member of A is also an element of B.
19. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Proper subset A⊂B
A is a subset of B and not equal to B.
Every member of A is also an element of B.
20. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Proper subset A⊂B
∀x:: A⊆B ∧ A≠B
A is a subset of B and not equal to B.
Every member of A is also an element of B.
26. Intersection A∩B
∀x:: x∈A ∧ x∈B
A∩B={ x | x∈A and x∈B }
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B A ∩ B = A.
31. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
32. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
33. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
34. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
35. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity
36. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∩ A’ = ∅
A ∪ A’ = U
Complement
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity
37. Similar to boolean algebra
a ∧ 1 = a
a ∨ 0 = a
Neutral elements
a ∧ 0 = 0
a ∨ 1 = 1
Zero elements
a ∧ a = a
a ∨ a = a
Idempotence
a ∧ ¬ a = 0
a ∨ ¬ a = 1
Negation
a ∨ b = b ∨ a
a ∧ b = b ∧ a
Commutativity
a ∧ (b ∧ c) = (a ∧ b) ∧ c
a ∨ (b ∨ c) = (a ∨ b) ∨ c
Associativity
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
Distributivity
38. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∩ A’ = ∅
A ∪ A’ = U
Complement
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity
39. A ∩ U = A A ∪ B = B ∪ A
A ∪ ∅ = A
A ∩ ∅ = ∅
A ∪ U = U
A ∩ A = A
A ∪ A = A
A ∩ A’ = ∅
A ∪ A’ = U
Neutral elements
Zero elements
Idempotence
Complement
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∩ B = B ∩ A
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(A ∩ B)’ = (A’) ∪ (B’)
(A ∪ B)’ = (A’) ∩ (B’)
Commutativity
Associativity
Distributivity
DeMorgan’s
40. A ⊆ A.
A ⊆ B ∧ B ⊆ A A = B.
A ⊆ B ∧ B ⊆ C A ⊆ C
Reflexivity
Anti-symmetry
Transitivity