LECTURE 2 1. Finitely Generated Abelian Groups We discuss the
... subgroups, each isomorphic to Z. Definition 1.7. Let C be a category and let X and Y be objects of C. A morphism f : X → Y is said to be a monomorphism when, for any object Z of C and any pair of morphisms i, j : Z → X, if f ◦ i = f ◦ j then i = j. Definition 1.8. Let C be a category and let X and Y ...
... subgroups, each isomorphic to Z. Definition 1.7. Let C be a category and let X and Y be objects of C. A morphism f : X → Y is said to be a monomorphism when, for any object Z of C and any pair of morphisms i, j : Z → X, if f ◦ i = f ◦ j then i = j. Definition 1.8. Let C be a category and let X and Y ...
IB Problems File
... The complex numbers z1 = 2 – 2i and z2 = 1 – i 3 are represented by the points A and B respectively on an Argand diagram. Given that O is the origin, (a) ...
... The complex numbers z1 = 2 – 2i and z2 = 1 – i 3 are represented by the points A and B respectively on an Argand diagram. Given that O is the origin, (a) ...
fundamental concepts of algebra - Department of Mathematical
... The set R of real numbers consists of all possible decimal expansions. We have shown that the rational numbers are precisely those real numbers with either terminating or repeating decimal expansions. As there are clearly decimal expansions that are not repeating (for example, 0.01011011101111 . . . ...
... The set R of real numbers consists of all possible decimal expansions. We have shown that the rational numbers are precisely those real numbers with either terminating or repeating decimal expansions. As there are clearly decimal expansions that are not repeating (for example, 0.01011011101111 . . . ...
Math 614, Fall 2015 Problem Set #1: Solutions 1. (a) Since every
... Q. If P and Q don’t have disjoint open neighborhoods, these have the FIP. A point in the intersection of all corresponds to a prime P0 such that both P and Q are in the closure of {P0 }, i.e. P0 ⊆ P ∩ Q.) 6. Call the K-subalgebra S. Then K ⊆ S ⊆ R. If S = K the result is clear. If not, let f ∈ S wit ...
... Q. If P and Q don’t have disjoint open neighborhoods, these have the FIP. A point in the intersection of all corresponds to a prime P0 such that both P and Q are in the closure of {P0 }, i.e. P0 ⊆ P ∩ Q.) 6. Call the K-subalgebra S. Then K ⊆ S ⊆ R. If S = K the result is clear. If not, let f ∈ S wit ...
Full text
... provides a partition of n with the given rank-vector. The one-one correspondence between the partitions of m and the matrices (4.1) establishes the following Tfieo/LCm: The number of partitions of n with the given rank-vector is the same as the number of partitions of m into at most k parts where m ...
... provides a partition of n with the given rank-vector. The one-one correspondence between the partitions of m and the matrices (4.1) establishes the following Tfieo/LCm: The number of partitions of n with the given rank-vector is the same as the number of partitions of m into at most k parts where m ...
Advanced Calculus
... The issue of convergence must not be ignored or casually assumed. The following example illustrates this: Consider the sequence ( xn ) defined by x1 1, xn 1 2 xn 1. Assuming the ‘convergence’ (actually wrong! The sequence is not convergent) with lim( xn ) x, we would obtain x 2x 1, so t ...
... The issue of convergence must not be ignored or casually assumed. The following example illustrates this: Consider the sequence ( xn ) defined by x1 1, xn 1 2 xn 1. Assuming the ‘convergence’ (actually wrong! The sequence is not convergent) with lim( xn ) x, we would obtain x 2x 1, so t ...
1-2 Note page
... Step 1: Multiply the first and last terms (6x)(-12x)=-72x2 Step 2: Find factors of -72 that will subtract or add to make +1 (coefficient of the middle term) 9x and -8x Step 3: Replace the middle term with 9x and -8x 6x2 + 9x – 8x – 12 Step 4: Factor out the Greatest Common Factor from the 1st and 2n ...
... Step 1: Multiply the first and last terms (6x)(-12x)=-72x2 Step 2: Find factors of -72 that will subtract or add to make +1 (coefficient of the middle term) 9x and -8x Step 3: Replace the middle term with 9x and -8x 6x2 + 9x – 8x – 12 Step 4: Factor out the Greatest Common Factor from the 1st and 2n ...
4.11. Proving and Disproving Set Statements. 4.11.1. Proof by
... Begin by letting x ∈ A , that is, we take x to be a particular but arbitrary element of A. Using the definitions, we prove that x ∈ B . As long as we use no special properties of the element x, we can conclude that A ⊆ B , which is what we wanted to prove. This method can be used to prove set equali ...
... Begin by letting x ∈ A , that is, we take x to be a particular but arbitrary element of A. Using the definitions, we prove that x ∈ B . As long as we use no special properties of the element x, we can conclude that A ⊆ B , which is what we wanted to prove. This method can be used to prove set equali ...
