GE013 Mathematics in the Modern World Lecture Notes

Vocabulary flashcards covering the nature of mathematics, logic, relations, functions, reasoning methods, and Polya's problem-solving steps based on the GE013 course notes.

Relation

A set of ordered pairs, often denoted as a subset of $A \times B$ where if an element $(x, y)$ exists, then $x$ corresponds to or depends on $y$.

Domain

The set of all $x$-values in a relation.

Image (or Range)

The set of all $y$-values in a relation.

Function

A relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

Logic

The science or study of how to evaluate arguments and reasoning.

Logical reasoning

The mathematics used to prove theorems.

Proposition (or statement)

A declarative sentence that is either true or false, but not both.

Truth Value

The truth and falsity of the proposition.

Propositional variable

A variable, such as $p$, $q$, or $r$, used to represent a proposition.

Compound proposition

A proposition composed of two or more simple propositions connected by logical connectives.

Simple (or Atomic) Proposition

A proposition that is not compound.

Conjunction

A compound proposition formed by combining simple propositions with the logical connective "and" ($\wedge$).

Disjunction

A compound proposition formed by combining simple propositions with the logical connective "or" ($\vee$).

Negation

A logical connective ($\sim$) used to state the opposite of a proposition, such as "not".

Conditional (or implication)

A compound proposition formed using the "if-then" ($\rightarrow$) logical connective.

Biconditional

A compound proposition formed using the "if and only if" ($\leftrightarrow$) logical connective.

Exclusive-or

The negation of a biconditional statement.

Predicate (or open statement)

A statement whose truth depends on the value of one or more variables, such as "$x$ is an even number".

Propositional function

A sentence $P(x)$ that becomes a statement only when the variable $x$ is given a particular value.

Inductive reasoning

The process of drawing a general conclusion from a repeated observation or limited sets of observations of specific examples.

Conjecture

The conclusion drawn by using inductive reasoning, or an unproven proposition that is believed to be true.

Counterexample

An example used to prove that a conjecture is false.

Deductive reasoning

The process of drawing a conclusion from a general case to specific examples, starting with a hypothesis and examining it to reach a specific conclusion.

Mathematical Intuition

A reliable mathematical belief without being formalized and proven directly, serving as the counterpart to rigorous geometry.

Mathematical Proof

An inferential argument for a mathematical statement that demonstrates it is always true in all possible cases.

Certainty

An essential defining attribute of mathematics and mathematical knowledge that, when correctly formulated, is beyond error and correction.

George Polya (1887-1985)

A mathematics educator who believed problem-solving skills can be taught and that it is the most efficient way of learning mathematical concepts.

Step 1: Understand the problem

The first step in Polya's four-step process, which involves identifying the goal, the knowns/unknowns, and the conditions of the problem.

Step 2: Devise a plan

The second step in Polya's process, involving strategies like working backward, using variables like $x$, writing equations, or looking for patterns.

Step 3: Carry out the plan

The third step in Polya's process, which involves implementing the strategy, being patient, keeping accurate records, and modifying the plan if it does not work.

Step 4: Look back

The final step in Polya's process, which involves rechecking computations, ensuring all conditions are met, and checking if the answer makes sense.