| SETS. SET OPERATIONS: UNION, INTERSECTION, DIFFERENCE, SYMMETRIC DIFFERENCE, CARTESIAN PRODUCT. THE SET OF SUBSETS OF A SET. PARTITIONS OF A SET. RELATIONS AND MAPS. IMAGES AND INVERSE IMAGES. INJECTIVE, SURJECTIVE, BIJECTIVE MAPS. COMPOSITION OF MAPS. THE INVERSE OF A BIJECTIVE MAP. EQUIVALENCE RELATIONS. EQUIVALENCE CLASSES. QUOTIENT SET. FUNDAMENTAL THEOREM. NATURAL NUMBERS AND INTEGER NUMBERS. THE PRINCIPLE OF MATHEMATICAL INDUCTION. DIVISIBILITY. EUCLIDEAN DIVISION. REPRESENTATION OF NATURAL NUMBERS IN A FIXED BASE. PRIME NUMBERS. THE FUNDAMENTAL THEOREM OF ARITHMETIC. EUCLID'S THEOREM ON THE EXISTENCE OF INFINITE PRIME NUMBERS. THE GREATEST COMMON DIVISOR AND THE LEAST COMMON MULTIPLE. EXTENDED EUCLIDEAN ALGORITHM. BEZOUT'S THEOREM. CONGRUENCES. LINEAR CONGRUENTIAL EQUATIONS. THE CHINESE REMAINDER THEOREM. MATRICES. MATRIX OPERATIONS: MATRIX SUM, SCALAR MULTIPLICATION, MATRIX PRODUCT, POWERS OF A MATRIX. TRANSPOSE OF A MATRIX. SCALING MATRIX. EQUIVALENT MATRICES. TRIANGULAR MATRIX. INVERTIBLE MATRICES. DETERMINANT OF A SQUARE MATRIX AND ITS REMARKABLE PROPERTIES. THE BINET'S THEOREM. CALCULATION OF THE INVERSE MATRIX OF AN INVERTIBLE MATRIX. THE RANK OF A MATRIX. ALGEBRAIC STRUCTURES. BINARY OPERATIONS IN A SET. MULTIPLICATION TABLE. STABLE SUBSETS AND INDUCED OPERATION. ASSOCIATIVE OPERATIONS. COMMUTATIVE OPERATIONS. IDENTITY ELEMENT. INVERTIBLE ELEMENTS. HOMOMORPHISMS. FUNDAMENTAL CONCEPTS ABOUT SEMIGROUPS, MONOIDS, GROUPS. THE GROUP OF UNITS OF A MONOID. MODULAR ARITHMETIC. FUNDAMENTAL CONCEPTS ABOUT RINGS, INTEGRAL DOMAINS, FIELDS. VECTOR SPACES. SUBSPACES AND GENERATORS. LINEAR DEPENDENCE, BASES AND DIMENSION. LINEAR APPLICATIONS. ISOMORPHIC VECTOR SPACES. SYSTEMS OF LINEAR EQUATIONS. BASIC CONCEPTS AND SOLVING METHODS: CRAMER, GAUSS-JORDAN, ROUCHE-CAPELLI.
DIAGONALIZATION OF A SQUARE MATRIX. EIGENVALUES AND EIGENVECTORS OF A SQUARE MATRIX. EIGENSPACES. SIMILAR MATRICES. DIAGONALIZABLE MATRICES. COMBINATORIAL CALCULUS. THE PRINCIPLE OF ADDITION. THE PRINCIPLE OF INCLUSION-EXCLUSION. THE PRINCIPLE OF MULTIPLICATION. FACTORIAL OF A NATURAL NUMBER. BINOMIAL COEFFICIENTS. DISPOSITIONS. DISPOSITIONS WITH REPETITIONS. PERMUTATIONS. PERMUTATIONS WITH REPETITIONS. COMBINATIONS. ORDER RELATIONS. MINIMAL ELEMENTS AND MAXIMAL ELEMENTS. MINIMUM AND MAXIMUM. UPPER BOUNDS AND LOWER BOUNDS. LEAST UPPER BOUND AND GREATEST LOWER BOUND. HASSE DIAGRAMS. TOTALLY ORDERED SETS. WELL-ORDERED SETS. SUBSETS OF AN ORDERED SET AND INDUCED ORDER. LATTICES. THE LATTICE OF SUBSETS OF A SET. THE LATTICE OF NON-NEGATIVE INTEGERS. SUBLATTICES. DISTRIBUTIVE LATTICES. LATTICES WITH COMPLEMENT. BOOLEN LATTICES. ELEMENTS OF ANALYTICAL GEOMETRY IN THE PLANE AND IN THE SPACE. APPLIED VECTORS AND RELATED OPERATIONS. AFFINE COORDINATES. PARAMETRIC AND CARTESIAN STRAIGHT LINE EQUATIONS IN THE PLANE AND IN THE SPACE. PARAMETRIC AND CARTESIAN PLANE EQUATIONS IN THE SPACE. PARALLELISM AND INCIDENCE CONDITIONS. |
