Mathematics and Logic - Skill and Concept Development

More Algebra Islands

Mathematical Recursion and Induction

The discussion of mathematical induction and recursion represents a standalone island of know-how combining logic and algebra. Mathematical induction and inductive definitions may be covered to explain or derive formulas for geometric sums and binomial coefficients. Mathematical induction may be woven into the further development of skills and concepts to add to the logical structure of mathematics as seen in this phase and the following ones.

Mathematical induction and recursion, starting with site logic chapters, is not a difficult topic to develop. Its coverage sets the stage for factorial notation, summation notation, product notation, binomial coefficients and summation formulas for sums: arithmetic, geometric and other. In any light development of calculus, that is calculus limited to polynomials and their ratios, mathematical induction would also play a role in obtaining rules for differentiation of polynomials directly and in composite functions where the last applied or outer function is polynomial.

Algebra Island: Natural Logarithm and Powers, etc.

The forward and backward use of the compound interest formula and encounters with radical and powers there-in provides a context or hook for the algebraic description of the properties of the natural logarithm, its inverse the exponential functions, and the expression of radicals and powers in terms of the latter inverse and the exponential powers. Deriving and providing those expression shows how and when radicals and powers can be computed.

Site algebra steps shows how the natural logarithm and its inverse or antilog, the natural exponential functions may be introduced numerically with an algebraic description of their fundamental properties. Following that, their properties may be used to show how to compute radical, powers and more exponentials using the natural logarithm and its inverse, the exponential function. Showing how to compute the latter with the aid of logarithms and its inverse indicate when they can computed as well. In essence, the site treatment of the foregoing provides unifies the otherwise disconnected treatments that I have seen in secondary mathematics.

Algebra Islands: Sets Concepts and Notation - A Balanced Approach

In mathematics for practical use, set concepts and operations of membership, union, intersection, and complements, an may help in the description of sets of numbers, in counting objects by dividing them among disjoint or overlapping set - think of Venn Diagrams; in discussing and illustraing logic - think again of Venn Diagrams; in the algebraic codification of probability theory in this phases, and in the algebraic description and codification of functions and their properties in the next phase. Before mastery of the algebraic statement of conditions to define subsets - see the safe set theory below, sets may be defined by a roster method - listing their members; and sets may introduced or depicted Geometrical as a set of points in the plane in as a set of points in a Venn Diagram.

Safe Set Theory

Everyday language uses the terms set, group and collection interchangeable. More formally, set concepts and operations in mathematics are codified in via the concepts and operations involving membership, union, intersection, complements, and what has been called set builder notation. The latter in fact would be better described as subset formation notation. In the eighteen hundreds, the too free or wild creation of sets led to self-reference paradoxes. To make set formation safer in the discussion of numbers, sets of numbers or their set theorectic equivalents were assumed. Then set formation was limited to forming subsets of a given set A, to set products sets

\[ A \times B = \{ [a,b] | a \in A, b \in B\} \]

that is sets of ordered pairs from two existing sets A and B; and to forming the power set 2^{A- the set of all subsets of an existing set A. The set operations of intersection, union and complement by taking placing a larger set may be cast as safe.}

Algebra Island: Probability Theory and Counting Methods

The coverage of probability is reserved to the end of phase 3. It requires coverage of function notation and review or coverage of set concepts and operations: membersbip, union, complement, intersection. The counting principles employed to the find the number of ways events or outcomes can be occur entail mastery of recursion in computation and in mathematical induction.

From home and elementary instruction, students may be familar with likelihood and chance. That is, Earlier use of likelihood in deciding whether or not to take a chance will be algebraically extended here into set-based probability theory. Mastery of probability and allied concepts of expected value may be presented as a base for taking or avoiding risks. The question of when to buy a lottery ticket or to visit a casino may be addressed in class.

Inverse Functions in Calculus or Before

Given the graph of a function y = f(x), one may use a vertical line method to find the value of f(x) from the graph, given the value of x in its domain. In that traditional, the graph of y versus x takes the y-axis to be vertical and the x-axis to be horizontal. To find or study the inverse of the function f, we may graph x versus y for the equation x = f(y). The table of values and graph of the latter is given by transpose the table of values and graph for y = f(x), provided we still keep the values of y along the vertical axis and those of x along the y-axis. On this graph of x =f(y), we may still try to apply the vertical line method to find y, given x. That works if the graph as a whole or a restricted portion of it, satisfies the vertical line rule. The foregoing provides a simple "dual" way to introduce and define a left inverse of a function. If we assume the transpose of a graph also transposes it tangent line, a formula for the derivative of the inverse is also suggested if not rigourous implied. For rigour, see the site treatment of this matter. The fact that the transpose [b,a] of a point [a,b] are reflections of each other may be mentioned in the foregoing, but is not critical. And in this development, the modern mathematics definion of a real-valued function y -f(x) as set of points in the plane need not appear. We simply take a formula for f(x) and it graph as two different computation rules for the same function.

Return to Page Top