Proof
Introduction: Proof is a fundamental part of mathematics that allows us to verify the truth of
mathematical statements. A proof is a logical argument that shows that a mathematical statement is
true. In this set of study notes, we will explore the structure of mathematical proofs and the different
methods of proof.
Structure of Mathematical Proof: The structure of a mathematical proof typically follows a few
basic steps:
1. Identify the given assumptions or premises of the argument.
2. Use logical deduction to apply mathematical rules and principles to derive new statements or
equations.
3. Continue applying logical deduction until the desired conclusion is reached.
4. State the conclusion and verify that it logically follows from the given assumptions and the
steps taken to derive it.
Methods of Proof: There are several methods of proof, including proof by deduction, proof by
exhaustion, disproof by counterexample, and proof by contradiction. We will discuss each of these
methods in more detail.
Proof by Deduction: Proof by deduction is the most common method of proof used in mathematics.
It involves starting with a set of given assumptions or premises and then using logical deduction to
derive new statements that logically follow from the assumptions. The proof ends with a conclusion
that is logically derived from the assumptions and intermediate steps.
Proof by Exhaustion: Proof by exhaustion involves proving a statement by showing that it holds for
all possible cases. This method is typically used when the number of cases is small and manageable.
For example, to prove that there are only five Platonic solids, we could enumerate all possible
regular convex polyhedra and show that only five meet the necessary criteria.
Disproof by Counterexample: Disproof by counterexample involves showing that a statement is
false by providing a specific example that contradicts it. This method is useful when trying to show
that a statement is not universally true. For example, the statement "All prime numbers are odd" can
be disproved by the counterexample of the number 2.
Proof by Contradiction: Proof by contradiction involves assuming the opposite of what you are
trying to prove and then showing that this leads to a contradiction. This method is often used when
it is difficult to directly prove a statement. For example, to prove that the square root of 2 is
irrational, we assume the opposite (i.e., that it is rational) and show that this leads to a contradiction.
Applications of Proof: Proof is used in many areas of mathematics, including number theory,
geometry, and calculus. It is also used in fields outside of mathematics, such as physics and
computer science, where mathematical models and algorithms are used to solve problems.
Conclusion: Proof is a crucial part of mathematics that allows us to verify the truth of mathematical
statements. The structure of mathematical proof involves starting with a set of given assumptions
and using logical deduction to derive new statements until a conclusion is reached. Different
methods of proof, such as proof by deduction, proof by exhaustion, disproof by counterexample,
and proof by contradiction, can be used to show the truth or falsehood of mathematical statements.
, Algebra and Functions
Introduction: Algebra and functions are essential parts of mathematics, and are used in a wide range
of fields from science and engineering to finance and economics. In this section, we will cover the
laws of indices, surds, quadratic functions, simultaneous equations, and linear and quadratic
inequalities.
1 Laws of indices for rational exponents:
The laws of indices are rules that describe how to simplify expressions with exponents. When the
exponents are rational numbers, the rules are:
a^m * a^n = a^(m+n) (a^m)^n = a^(mn) a^(-m) = 1/a^m a^(m/n) = nth root of a^m
For example, simplify the expression (2^3 * 3^2)^1/2:
(2^3 * 3^2)^1/2 = (8 * 9)^1/2 = 24^1/2
2 Surds:
A surd is an expression containing a root of a positive integer that cannot be simplified to a rational
number. To manipulate surds, we use the following rules:
sqrt(a) * sqrt(b) = sqrt(ab) sqrt(a/b) = sqrt(a) / sqrt(b)
For example, simplify the expression (2sqrt(3) + sqrt(12)) / (sqrt(3)):
(2sqrt(3) + sqrt(12)) / (sqrt(3)) = 2sqrt(3) / sqrt(3) + sqrt(12) / sqrt(3) = 2 + 2sqrt(3)
3 Quadratic functions:
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
The graph of a quadratic function is a parabola, and the discriminant of a quadratic function is given
by b^2 - 4ac. If the discriminant is positive, the quadratic function has two real roots, if the
discriminant is zero, the quadratic function has one real root, and if the discriminant is negative, the
quadratic function has no real roots.
For example, find the roots of the quadratic equation 3x^2 - 4x - 1 = 0:
Using the quadratic formula, we have:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
x = (4 +/- sqrt(16 + 12)) / 6 = (4 +/- 2sqrt(7)) / 6
Therefore, the roots of the quadratic equation are (4 + 2sqrt(7)) / 6 and (4 - 2sqrt(7)) / 6.
4 Simultaneous equations:
Simultaneous equations are equations that are solved together. To solve simultaneous equations in
two variables by elimination, we eliminate one of the variables by adding or subtracting the
equations. To solve simultaneous equations in two variables by substitution, we substitute one of the
variables in terms of the other variable.
For example, solve the simultaneous equations:
x + y = 3 x^2 + y^2 = 13
From the first equation, we have y = 3 - x. Substituting this in the second equation, we get:
x^2 + (3 - x)^2 = 13
Expanding and simplifying, we get:
2x^2 - 6x + 2 = 0
Introduction: Proof is a fundamental part of mathematics that allows us to verify the truth of
mathematical statements. A proof is a logical argument that shows that a mathematical statement is
true. In this set of study notes, we will explore the structure of mathematical proofs and the different
methods of proof.
Structure of Mathematical Proof: The structure of a mathematical proof typically follows a few
basic steps:
1. Identify the given assumptions or premises of the argument.
2. Use logical deduction to apply mathematical rules and principles to derive new statements or
equations.
3. Continue applying logical deduction until the desired conclusion is reached.
4. State the conclusion and verify that it logically follows from the given assumptions and the
steps taken to derive it.
Methods of Proof: There are several methods of proof, including proof by deduction, proof by
exhaustion, disproof by counterexample, and proof by contradiction. We will discuss each of these
methods in more detail.
Proof by Deduction: Proof by deduction is the most common method of proof used in mathematics.
It involves starting with a set of given assumptions or premises and then using logical deduction to
derive new statements that logically follow from the assumptions. The proof ends with a conclusion
that is logically derived from the assumptions and intermediate steps.
Proof by Exhaustion: Proof by exhaustion involves proving a statement by showing that it holds for
all possible cases. This method is typically used when the number of cases is small and manageable.
For example, to prove that there are only five Platonic solids, we could enumerate all possible
regular convex polyhedra and show that only five meet the necessary criteria.
Disproof by Counterexample: Disproof by counterexample involves showing that a statement is
false by providing a specific example that contradicts it. This method is useful when trying to show
that a statement is not universally true. For example, the statement "All prime numbers are odd" can
be disproved by the counterexample of the number 2.
Proof by Contradiction: Proof by contradiction involves assuming the opposite of what you are
trying to prove and then showing that this leads to a contradiction. This method is often used when
it is difficult to directly prove a statement. For example, to prove that the square root of 2 is
irrational, we assume the opposite (i.e., that it is rational) and show that this leads to a contradiction.
Applications of Proof: Proof is used in many areas of mathematics, including number theory,
geometry, and calculus. It is also used in fields outside of mathematics, such as physics and
computer science, where mathematical models and algorithms are used to solve problems.
Conclusion: Proof is a crucial part of mathematics that allows us to verify the truth of mathematical
statements. The structure of mathematical proof involves starting with a set of given assumptions
and using logical deduction to derive new statements until a conclusion is reached. Different
methods of proof, such as proof by deduction, proof by exhaustion, disproof by counterexample,
and proof by contradiction, can be used to show the truth or falsehood of mathematical statements.
, Algebra and Functions
Introduction: Algebra and functions are essential parts of mathematics, and are used in a wide range
of fields from science and engineering to finance and economics. In this section, we will cover the
laws of indices, surds, quadratic functions, simultaneous equations, and linear and quadratic
inequalities.
1 Laws of indices for rational exponents:
The laws of indices are rules that describe how to simplify expressions with exponents. When the
exponents are rational numbers, the rules are:
a^m * a^n = a^(m+n) (a^m)^n = a^(mn) a^(-m) = 1/a^m a^(m/n) = nth root of a^m
For example, simplify the expression (2^3 * 3^2)^1/2:
(2^3 * 3^2)^1/2 = (8 * 9)^1/2 = 24^1/2
2 Surds:
A surd is an expression containing a root of a positive integer that cannot be simplified to a rational
number. To manipulate surds, we use the following rules:
sqrt(a) * sqrt(b) = sqrt(ab) sqrt(a/b) = sqrt(a) / sqrt(b)
For example, simplify the expression (2sqrt(3) + sqrt(12)) / (sqrt(3)):
(2sqrt(3) + sqrt(12)) / (sqrt(3)) = 2sqrt(3) / sqrt(3) + sqrt(12) / sqrt(3) = 2 + 2sqrt(3)
3 Quadratic functions:
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
The graph of a quadratic function is a parabola, and the discriminant of a quadratic function is given
by b^2 - 4ac. If the discriminant is positive, the quadratic function has two real roots, if the
discriminant is zero, the quadratic function has one real root, and if the discriminant is negative, the
quadratic function has no real roots.
For example, find the roots of the quadratic equation 3x^2 - 4x - 1 = 0:
Using the quadratic formula, we have:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
x = (4 +/- sqrt(16 + 12)) / 6 = (4 +/- 2sqrt(7)) / 6
Therefore, the roots of the quadratic equation are (4 + 2sqrt(7)) / 6 and (4 - 2sqrt(7)) / 6.
4 Simultaneous equations:
Simultaneous equations are equations that are solved together. To solve simultaneous equations in
two variables by elimination, we eliminate one of the variables by adding or subtracting the
equations. To solve simultaneous equations in two variables by substitution, we substitute one of the
variables in terms of the other variable.
For example, solve the simultaneous equations:
x + y = 3 x^2 + y^2 = 13
From the first equation, we have y = 3 - x. Substituting this in the second equation, we get:
x^2 + (3 - x)^2 = 13
Expanding and simplifying, we get:
2x^2 - 6x + 2 = 0
