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Proof by Deduction

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
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- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
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- Limit Laws
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- Limit of a Sequence
- Limits
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- Limits of a Function
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- Linear Differential Equation
- Linear Functions
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- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
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- Polar Coordinates
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- Population Change
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- Riemann Sum
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- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
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- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
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- Probability and Statistics
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- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
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- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
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- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
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- Approximation and Estimation
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- Arithmetic Sequences
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- Combination of Functions
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- Direct and Inverse proportions
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- Disproof by Counterexample
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- Equations and Identities
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- Estimation in Real Life
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- Even Functions
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- Finding Rational Zeros
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- Forms of Quadratic Functions
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- Fractions and Factors
- Fractions in Expressions and Equations
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- Function Basics
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- Functions
- Fundamental Counting Principle
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- Generating Terms of a Sequence
- Geometric Sequence
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- Graphs
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- Graphs of Common Functions
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- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
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- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
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- Integration of Hyperbolic Functions
- Interest
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- Inverse Matrices
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- Iterative Methods
- Law of Cosines in Algebra
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- Laws of Logs
- Limits of Accuracy
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- Location of Roots
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- Notation
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- Permutations and Combinations
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- Points Lines and Planes
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- Problem-solving Models and Strategies
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- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
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‘If today is a weekend, then tomorrow must be a weekday.’

This statement can either be true or false, which makes it perfect for proof by deduction. You can split this statement into two parts: Today is a weekend (A); tomorrow must be a weekday (B). Mathematically, you can write it as:

, where is the symbol meaning ‘implies’.

In Proof by Deduction, the truth of the statement is based on the truth of each part of the statement (A; B) and the strength of the logic connecting each part.

Statement A: ‘if today is a weekend’ gives us two answers, Saturday and Sunday, as these are the only two days of the weekend.

We then use our answers for statement A and statement B to test the logic of the main statement.

If today is Saturday, then tomorrow is a Sunday. Thus, the concluding statement is false. However, if today is Sunday, tomorrow is Monday, and the concluding statement is true.

Therefore, the logic of the concluding statement depends on statement A and is weak as a result.

In Maths, the concluding statements tend to have more conclusive answers (because numbers don’t lie!). To prove a mathematical conclusion ( **conjecture** ) by proof of deduction, you need **strong **mathematical axioms and logic.

Mathematical axioms are the mathematical concepts underlining the concluding statement.

To solve a Proof by Deduction question, you must:

- Consider the logic of the conjecture.
- Express the axiom as a mathematical expression where possible.
- Solving through to see if the logic applies to the conjecture.
- Making a concluding statement about the truth of the conjuncture.

Although most of these algebraic rules will be familiar to you, it is good to stay familiar with them as expressing axioms as a mathematical expression sometimes requires some creativity using these rules.

*n * stands in for any number.

- To express
*n*is a multiple of A, you can write as*An*

Express *n as *a multiple of 12 mathematically.

A is 12. Therefore, the answer is 12n

- To express consecutive numbers, you can start with
*n*(or any other starting point) and add one each time to get*n + 1, n + 2,*etc*.*

Express the next two consecutive numbers after

To get the following consecutive numbers, you add 1 to each consecutive number. Therefore, the first term is , the second term is , the third term is .

- To express consecutive even numbers, you can start with the consecutive numbers:
*n, n + 1, n + 2*. You then multiply each term by 2 as all even numbers are multiples of 2. Therefore the consecutive even terms are*2 (n), 2 (n + 1), 2 (n + 2)*which can be simplified to*2n, 2n + 2, 2n + 4 etc.* - Expressing consecutive odd numbers is a little bit more complicated than expressing consecutive even numbers as odd numbers are not part of a multiple. However, they are defined by not being a multiple of two; therefore, all the gaps in the consecutive even numbers will make up the consecutive odd numbers.

Consecutive even numbers | 2n | 2n + 2 | 2n + 4 | |||

Consecutive odd numbers | 2n + 1 | 2n + 3 | 2n + 5 |

We will now go through a few examples to show how you answer questions like these.

Prove the sum of two consecutive numbers is equivalent to the difference between two consecutive numbers squared.

As described above, you can algebraically express two consecutive numbers as *n, n + 1* .

The sum of two consecutive numbers is therefore

To find the difference between two consecutive numbers squared, you first have to square each consecutive number to get and .

Expanding out and simplifying the squares gives you:

Therefore the difference between two consecutive numbers squared is

To finish off the question, you must write a concluding statement: * The sum of two consecutive numbers and the difference between two consecutive numbers squared is equal to each other as they are both equal to 2n + 1.*

Prove the answer to the equation is always positive.

As you only want one variable of *x*, you need to complete the square with the equation.

1. First, you halve b (8) and substitute it into your new equation:

2. You then expand out to find your constant outside the bracket. You need +20 to make the new equation match the same as the equation, so you need to +4. Therefore, the answer is

As always, you need a concluding statement to explain the maths: *Regardless of the value of x, by squaring it and adding 4, the value of the equation will always be positive.*

- Proof by deduction uses mathematical axioms and logic to prove or disprove a conjecture.
- You can express several axioms algebraically, like even and odd consecutive numbers.

1. Consider the logic of the conjecture.

2. Express the axiom as a mathematical expression where possible.

3. Solve through to see if the logic applies to the conjecture.

4. Make a concluding statement about the truth of the conjecture.

Deductive reasoning was introduced by Aristotle.

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