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Elements of Set Theory and Mathematical logic: Course Contents (MAT 102)

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects

Course Outline

Set theory:

Sets, subsets, operations with sets:

union,intersection, difference, symmetric difference,

Cartesian product and disjoint union.

Functions: graph of a function.

Composition; injections,

surjections, bijections, inverse function.

Computing cardinals:

Cardinality of Cartesian product, union.

Cardinality of all functions from a set to another  set.  

Cardinality  of  all  injective,  surjective 

bijective functions from a set to another set.

Infinite sets, finite sets.

Countable sets,

properties, examples (Z, Q).

R is not countable. R, RxR, RxRxR have the same cardinal.

Operations with cardinal numbers.

Cantor-Bernstein theorem.

Relations:

Equivalence relations,

partitions, quotient set; examples,

parallelism, similarity of triangles.

Order relations, min, max, inf, sup; linear order.

Examples: N, Z, R, P(A). Well ordered sets and induction.

Inductively ordered sets and Zorn’s lemma.

Mathematical logic:

Propositional Calculus.

Truth tables. Predicate Calculus.

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