Fundamental Tactics for Solving Problems

1. Fundamental Tactics for Solving Problems Prepared by AlmiraCattleya L. Cariño

2. Fundamental Tactics for Solving Problems • What is symmetry? • Symmetry in mathematics

3. Fundamental Tactics for Solving Problems • TACTICS are broadly applicable mathematical methods that often simplify problems. • STRATEGY alone rarely solves problems; we need the more focused power of tactics (and often highly specialized tools as well) to finish the job. • Most of the strategic ideas are plain common sense. In tactical ideas in this easy t use, are less “natural” as few people would think of them. Lets return to our mountaineering analogy for a moment. An important climbing tactics is the rather non- obvious idea ( meant to be taken literally ).

4. What is Symmetry? • Symmetry is a model topic for study in school. It is embedded in reality, it is conceptually simple for younger pupils, and concrete examples around. • symmetry also provides various opportunities for students to enjoy learning mathematics. It helps students visualize different geometry concepts and connect learning with their real-life experience. • Symmetry involves finding or imposing order in a concrete way for examples by reflections.

5. Symmetry in mathematics • Symmetric functions • In algebra • In geometry

6. Symmetric functions In the case of symmetric functions, the value of the output is invariant under permutations of variables. From the form of an equation one may observe that certain permutations of the unknowns result in an equivalent equation. In that case the set of solutions is invariant under any permutation of the unknowns in the group generated by the aforementioned permutations. For example • (a − b)(b − c)(c − a) = 10, for any solution (a,b,c), permutations (abc) and (acb) can be applied giving additional solutions (b, c, a) and (c, a, b). • a2c + 3ab + b2c remains unchanged under interchanging of a and b.

7. In algebra • A symmetric matrix, seen as a symmetric function of the row- and column number, is an example. The second order partial derivatives of a suitably smooth function, seen as a function of the two indexes, is another example. See also symmetry of second derivatives. • A relation is symmetric if and only if the corresponding boolean-valued function is a symmetric function. • A binary operation is commutative if the operator, as function of two variables, is a symmetric function. Symmetric operators on sets include the union, intersection, and symmetric difference.

8. In geometry • By considering the coordinate space we can consider the symmetry in geometric terms. In the case of three variables we can use e.g. Schoenflies notation for symmetries in 3D. In the example the solution set is geometrically in coordinate space at least of symmetry type C3. If all permutations were allowed this would be C3v. If only two unknowns could be interchanged this would be Cs. • In fact, prior to the 20th century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20th century that the current abstract definition of a group without any reference to group actions was used instead.