Mathematical Argument, Language, and Proof
Introduction: In mathematics, a mathematical argument is a set of statements or propositions,
supported by valid reasoning or evidence, leading to a conclusion. The language and symbols used
in mathematical arguments are precise and have specific meanings. This ensures that
communication is clear and unambiguous. Mathematical arguments are used to establish the
validity of mathematical statements, and proofs are used to establish the truth of mathematical
theorems.
1. Construct and present mathematical arguments through appropriate use of diagrams; sketching
graphs; logical deduction; precise statements involving correct use of symbols and connecting
language, including: constant, coefficient, expression, equation, function, identity, index, term,
variable.
1.1. Diagrams: Diagrams are useful tools for representing mathematical concepts visually. They can
be used to illustrate the relationships between mathematical objects, such as sets, functions, and
graphs. Diagrams can also be used to show geometric constructions and geometric properties of
figures. It is important to label and annotate diagrams clearly and accurately.
1.2 Graphs: Graphs are another visual tool used in mathematics to represent functions and
relationships between variables. They can be used to analyze data and make predictions. When
sketching graphs, it is important to accurately plot points and label axes with appropriate scales and
units.
1.3 Logical deduction: Logical deduction involves using reasoning to draw conclusions from given
premises. In mathematical arguments, deductive reasoning is used to establish the truth of
mathematical statements. It is important to use valid deductive arguments, which are arguments that
are logically valid, and the premises lead to a conclusion that follows logically from the premises.
1.4 Precise statements involving correct use of symbols and connecting language: Precise
statements involve using correct symbols and language to communicate mathematical ideas clearly
and unambiguously. Common symbols used in mathematics include +, −, ×, ÷, =, >, <, and √.
Connecting language, such as “there exists,” “for all,” and “implies,” are used to convey logical
relationships between mathematical objects.
1.5 Constant, coefficient, expression, equation, function, identity, index, term, variable: These terms
are commonly used in mathematics and have specific meanings. A constant is a fixed value, such as
π or e. A coefficient is a number that multiplies a variable in an algebraic expression. An expression
is a combination of numbers, variables, and operations, such as x² + 3x − 2. An equation is a
statement that equates two expressions, such as x² + 3x − 2 = 0. A function is a relation between two
sets that assigns a unique output to each input. An identity is an equation that is true for all values of
the variables. An index is a small number or letter used to identify the position of an element in a
sequence or matrix. A term is a single element in an algebraic expression. A variable is a symbol
used to represent a quantity that can vary.
2 Understand and use mathematical language and syntax as set out in the content.
2.1 Mathematical language: Mathematical language is a precise and formal language used to
communicate mathematical ideas. It is important to use mathematical language accurately to avoid
ambiguity and misinterpretation. Common examples of mathematical language include variables,
equations, inequalities, and functions.
2.2 Mathematical syntax: Mathematical syntax refers to the rules governing the construction of
mathematical statements. These rules include using appropriate symbols and notation, and
following standard conventions for order of operations, grouping symbols, and exponents.
3 Understand and use language and symbols associated with set theory, as set out in the appendices.
Apply to solutions of inequalities and probability.
, 3.1 Language and symbols associated with set theory: Set theory is a branch of mathematics that
deals with sets, which are collections of objects. In mathematical arguments, it is common to use set
notation and symbols, such as ∈ (element of), ⊆ (subset), and ∪ (union), to represent relationships
between sets. Understanding set theory and its associated language and symbols is important for
many areas of mathematics.
3.2 Solutions of inequalities: Inequalities are mathematical statements that express a relationship
between two quantities, often using symbols such as >, <, and ≤. Inequalities can be solved by
manipulating the statement algebraically to isolate the variable, or by graphing the solution set on a
number line. It is important to use appropriate symbols and notation when solving inequalities to
avoid confusion and errors.
3.3 Probability: Probability is a branch of mathematics that deals with the study of randomness and
uncertainty. In mathematical arguments involving probability, it is common to use symbols and
notation such as P(A) (probability of event A), P(A ∩ B) (probability of the intersection of events A
and B), and P(A | B) (probability of event A given that event B has occurred). Understanding
probability notation and its associated language and symbols is important for many areas of
mathematics, including statistics and data analysis.
4 Understand and use the definition of a function; domain and range of functions.
4.1 Definition of a function: A function is a mathematical object that assigns a unique output to each
input. The input is called the domain, and the output is called the range. A function can be
represented algebraically, graphically, or verbally. It is important to understand the definition of a
function and how to identify a function from a given set of data or information.
4.2 Domain and range of functions: The domain of a function is the set of all possible input values,
and the range is the set of all possible output values. The domain and range of a function can be
determined by examining the definition of the function or by analyzing the graph of the function. It
is important to understand the domain and range of a function, as they can affect the behavior and
properties of the function.
5 Comprehend and critique mathematical arguments, proofs and justifications of methods and
formulae, including those relating to applications of mathematics.
5.1 Comprehend mathematical arguments, proofs, and justifications: Comprehending mathematical
arguments, proofs, and justifications involves understanding the reasoning and evidence used to
support a mathematical statement or theorem. It is important to be able to identify the assumptions
and logical steps taken in a mathematical argument, and to understand how they lead to the
conclusion.
5.2 Critique mathematical arguments, proofs, and justifications: Critiquing mathematical arguments,
proofs, and justifications involves evaluating the validity and soundness of a mathematical
statement or theorem. It is important to be able to identify errors or flaws in mathematical
arguments, and to provide alternative approaches or counterexamples when necessary. Critiquing
mathematical arguments is a crucial skill in mathematics, as it promotes a deeper understanding and
appreciation of the subject.
Conclusion: Mathematical argument, language, and proof are essential skills for any student of
mathematics. By understanding how to construct and present mathematical arguments, use
mathematical language and symbols accurately, and comprehend and critique mathematical
arguments and proofs, students will be better equipped to succeed in the study and application of
mathematics.
