Practical (elementary) Logic and statements Sets Countability Relations: equivalence and order Fields R as a completion of Q
MATH 464
Marco Bertola
September 9, 2026
,Practical (elementary) Logic and statements Sets Countability Relations: equivalence and order Fields R as a completion of Q
Math 464: class 1 Review
1 Practical (elementary) Logic and statements
2 Sets
Complement, union, intersection
Collections of sets
Algebras of sets
Cartesian products
3 Countability
4 Relations: equivalence and order
Example: construction of Z from N and Q from Z
5 Fields
6 R as a completion of Q
, Practical (elementary) Logic and statements Sets Countability Relations: equivalence and order Fields R as a completion of Q
Common symbols
P,Q belongs to
R, S does not belong to
Ă, Ą set inclusions
Ĺ, Ľ strict set inclusions
D there exists
@ for all
D! there is a unique
: or s.t. or | such that (satisfying a property that follows)
X, Y Intersection, union
H empty set
à or Ac complement of set A in a qualifying ambient set
ñ P implies Q, or P is sufficient for Q or Q is necessary for P
ô logical equivalence (if and only if)
_ logical "or" (not exclusive)
^ logical "and"
def
:“ or ” or “ symbol of "definition" (the rhs defines the lhs)
We assume a working knowledge (from Math 364) of the use of quantifiers @, D and Boolean operators, and (^),
"or" (_), "not" (␣) their negation rules. We will use them in a semi-formal way.
For example: saying that one set X is a subset of Y (X Ă Y ) means @x P X, x P Y or also
x P X ñ x P Y . Note that inclusion is not strict. To disambiguate one can use Ĺ. Another example: X Ĺ Y
means that X Ă Y and ␣pY Ă Xq, or equivalently, p@x P X, x P Y q ^ pDy P Y : y R Xq.
MATH 464
Marco Bertola
September 9, 2026
,Practical (elementary) Logic and statements Sets Countability Relations: equivalence and order Fields R as a completion of Q
Math 464: class 1 Review
1 Practical (elementary) Logic and statements
2 Sets
Complement, union, intersection
Collections of sets
Algebras of sets
Cartesian products
3 Countability
4 Relations: equivalence and order
Example: construction of Z from N and Q from Z
5 Fields
6 R as a completion of Q
, Practical (elementary) Logic and statements Sets Countability Relations: equivalence and order Fields R as a completion of Q
Common symbols
P,Q belongs to
R, S does not belong to
Ă, Ą set inclusions
Ĺ, Ľ strict set inclusions
D there exists
@ for all
D! there is a unique
: or s.t. or | such that (satisfying a property that follows)
X, Y Intersection, union
H empty set
à or Ac complement of set A in a qualifying ambient set
ñ P implies Q, or P is sufficient for Q or Q is necessary for P
ô logical equivalence (if and only if)
_ logical "or" (not exclusive)
^ logical "and"
def
:“ or ” or “ symbol of "definition" (the rhs defines the lhs)
We assume a working knowledge (from Math 364) of the use of quantifiers @, D and Boolean operators, and (^),
"or" (_), "not" (␣) their negation rules. We will use them in a semi-formal way.
For example: saying that one set X is a subset of Y (X Ă Y ) means @x P X, x P Y or also
x P X ñ x P Y . Note that inclusion is not strict. To disambiguate one can use Ĺ. Another example: X Ĺ Y
means that X Ă Y and ␣pY Ă Xq, or equivalently, p@x P X, x P Y q ^ pDy P Y : y R Xq.
