Reviewer (Lesson 1-3)

• Mathematics is a branch of science that deals with numbers and their operations, involving calculation, computation, and solving problems.
• Mathematics helps us to organize and systemize our ideas about patterns, allowing us to admire and enjoy these patterns, and use them to infer some of the underlying principles that govern the world of nature.
• Mathematics is a study of patterns and relationships, a way of thinking, an art, a language, and a tool.
• A pattern is an arrangement which helps observers anticipate what they might see or what happens next.
• The human mind is programmed to make sense of data or to bring order where there is disorder, seeking to discover relationships and connections between seemingly unrelated bits of information.
• Patterns in nature are visible regularities of form found in the natural world, which recur in different contexts and can sometimes be modeled mathematically.
• Symmetry occurs when there is congruence in dimensions, due proportions and arrangement, providing a sense of harmony and balance.
• A subset is a collection of elements that is contained within another collection.
• Contradiction refers to all false statements.
• Contingency refers to both true and false statements.
• Logic puzzles can be solved using deductive reasoning and a chart that enables us to display the given information in a visual manner.
• Tautology refers to all true statements.
• Cartesian product is the product of two sets, represented as A x B.
• A universal set is the set of all elements.
• Intersect is the process of intersecting two or more sets, represented as ∩.
• Union is the process of combining two or more sets, represented as ∪.
• An empty set or null set is represented as ∅ or {}.
• Difference is the process of subtracting two sets, represented as A - B.
• Completement is the complement of a set, represented as A’.
• Bilateral or reflection symmetry is the simplest kind of symmetry, one of the most common kinds of symmetry that we see in the natural world, and can also be called mirror symmetry because an object with this symmetry looks unchanged if a mirror passes through its middle.
• If a shape can be folded in half so that one half fits exactly on top of the other, then we say that the shapes are symmetric.
• Bilateral-symmetric objects have at least one line or axis of symmetry, which may be in any direction.
• Radial Symmetry is rotational symmetry around a fixed point known as the center, which can be found both in natural and human made objects.
• Isometries of the Plane Transformation is a process which shifts points of the plane to possibly new locations in the plane.
• Translation (or slide) in Isometries of the Plane Transformation moves a shape in a given direction by sliding it up,down, sideways, or diagonally.
• Counterexamples are examples which can negate our conjectures.
• Only one counter example can prove that our conjecture is false.
• The cardinal number of a finite set A is the unique counting number n such that the elements of A are in one-to-one correspondence with the elements of the set.
• If A and B are disjoint then the intersection of A and B is the null set, i.e A ∩ B = ∅.
• Disjoint sets A and B, if x ∈ A, then x ∉ B and if x ∈ B, then x ∉ A.
• The set product or Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) where a is in A and b is in B, denoted by A × B.
• The complement of A is defined to be the set of elements of U which are not in A, denoted by A’.
• The cardinal number of A is denoted by the symbol n(A) and is referred to as the cardinal number of A.
• There are two major types of reasoning: inductive and deductive.
• Logically speaking, we cannot prove a general statement from a number of specific examples unless there are only finitely many examples and we can exhaust them.
• Human beings are said to be rational creatures because we use reasoning to come up with sound decisions that we have to make everyday.
• All the sets under investigation will likely be subsets of a fixed set, which is denoted by U.
• The union of sets A and B is defined to be the set of elements that belong to A or to B, denoted by A ∪ B.
• Conjecture is a conclusion made using inductive reasoning.
• The intersection of two sets A and B is the set of all elements that belong to both A and B, denoted by A ∩ B.