Mathematical Reasoning, Logic, and Applied Mathematics Flashcards

Mathematical Reasoning

Analyzing information and forming logical arguments to determine whether a conclusion follows correctly from given facts.

Proposition

A statement that is either true or false.

Negation

The logical statement $\neg p$, meaning "not $p$".

Conjunction

The logical statement $p \wedge q$, meaning "$p$ and $q$".

Disjunction

The logical statement $p \vee q$, meaning "$p$ or $q$".

Conditional

The logical statement $p \rightarrow q$, meaning "if $p$, then $q$", which is false only when $p$ is true and $q$ is false.

Biconditional

The logical statement $p \leftrightarrow q$, meaning "$p$ if and only if $q$".

Converse

The conditional statement $q \rightarrow p$, derived from $p \rightarrow q$.

Inverse

The conditional statement $\neg p \rightarrow \neg q$, derived from $p \rightarrow q$.

Contrapositive

The conditional statement $\neg q \rightarrow \neg p$, derived from $p \rightarrow q$.

Inductive Reasoning

Reasoning that forms a general conclusion from specific examples.

Deductive Reasoning

Reasoning that applies a general principle to reach a specific conclusion.

Conjecture

A conclusion based on observed patterns.

Counterexample

A single example that disproves a universal claim.

Valid Argument

An argument in which the conclusion follows logically from the premises.

Invalid Argument

An argument in which the conclusion does not necessarily follow from the premises.

Polya's Four-Step Problem-Solving Strategy

A systematic method to solve unfamiliar problems consisting of four steps: 1. Understand the problem, 2. Devise a plan, 3. Carry out the plan, and 4. Review the solution.

Sequence

An ordered list of numbers or objects that follows a rule.

Arithmetic Sequence

A sequence that has a constant difference $d$, with general term $a_n = a_1 + (n - 1)d$.

Geometric Sequence

A sequence that has a constant ratio $r$, with general term $a_n = a_1 r^{n-1}$.

Arithmetic Sum

The sum of the first $n$ terms of an arithmetic sequence, given by $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$.

Geometric Sum

The sum of the first $n$ terms of a geometric sequence, given by $S_n = \frac{a_1(r^n - 1)}{r - 1}$ for $r \neq 1$.

Set

A collection of clearly defined objects.

Union

The set operation $A \cup B$, which includes elements in set $A$, set $B$, or both.

Intersection

The set operation $A \cap B$, which includes elements common to both set $A$ and set $B$.

Difference (Set)

The set operation $A - B$, which includes elements in set $A$ but not in set $B$.

Complement

The set operation $A'$, which includes elements not in set $A$.

Empty Set

A set containing no elements, denoted as $\emptyset$.

Universal Set

The set $U$, representing all elements under consideration.

Cardinality

The number of elements in set $A$, represented as $n(A)$.

Addition Rule for Sets

Formula for finding the cardinality of the union of two sets: $$n(A \cup B) = n(A) + n(B) -