Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 → Mathematical Reasoning → Proofs ↓


Indirect Proof


In mathematics and logic, proving things is fundamental. Proofs not only help us know if something is true or false, but they also help us understand why it is so. There are many ways to prove something, and one very interesting way is called indirect proof.

What is indirect evidence?

Indirect proof is a method of proving a statement in which it is assumed that the statement to be proved is false and then it is shown that this assumption leads to a contradiction. Since the assumption that the statement is false leads to inconsistency, we conclude that the statement must be true. This method is sometimes known as proof by contradiction.

Basic structure of indirect proof

  • Start with the assumption that the statement we want to prove is false.
  • Logically draw out the implications of this assumption.
  • Show that at least one of these implications is impossible or leads to a contradiction.
  • Conclude that the original statement must be true because assuming it to be false results in a contradiction.

Advantages of using indirect evidence

The indirect proof method is especially useful when:

  • Direct evidence is very complex or impossible to obtain easily.
  • The problem naturally sets up as a contradiction.
  • This allows for a variety of viewpoints and perspectives.

Simple example

Simple mathematical example

Let's start with a simple logical statement:

Statement: Suppose an integer n is odd. Prove that n 2 is odd.

Evidence:

For the sake of contradiction, suppose that n 2 is even.

  • If n 2 is even, then it can be written as 2k for some integer k.
  • Now, since n 2 = 2k, n itself must be even. This is because the square of an odd number is odd, so this assumption implies that n is even.
  • However, this is a contradiction since we initially assumed that n is odd.

Thus, our assumption that n 2 is even will be false. Therefore, n 2 is odd.

Let's look at this simple proof:

Assume that n^2 is even n is even (contradiction) n must be odd, n^2 is odd

Examples in geometry

Consider a classic statement of geometry:

Statement: Prove that the square root of 2 is irrational.

Evidence:

For the sake of contradiction assume that sqrt(2) is a rational number.

  • If sqrt(2) is rational, it can be expressed as a fraction a/b where a and b are integers with no common factors other than 1 (that is, a/b is in its simplest form).
  • Then, sqrt(2) = a/b implies 2 = a 2 /b 2 or a 2 = 2b 2.
  • This implies that a2 is even, since 2 times any integer is even.
  • If a2 is even, then a must also be even (since the square of an odd number is odd).
  • Let a = 2c for some integer c. Then, (2c) 2 = 2b 2 or 4c 2 = 2b 2 which simplifies to 2c 2 = b 2.
  • Thus, b 2 is even, which means that b must also be even.

If a and b are both even, then their common factor will be 2, which contradicts our assumption that a/b is in simplest form.

Therefore, the assumption that sqrt(2) is rational leads to a contradiction, which means that sqrt(2) must be irrational.

Using indirect evidence in real-life situations

Indirect proofs are not just limited to our math classes; they have applications in real life as well. Consider the scenario where we want to demonstrate the unreliability of a particular system design. By assuming that a flawed design works flawlessly, and then showing how this assumption leads to operational errors or inconsistencies, we effectively prove flaws by contradiction.

Reasoning with a logical position

Let us consider a logical example to understand indirect proof better:

Scenario: You have a roommate, and you've noticed that some cookies are missing from your cookie jar. Suspect that either your roommate or you ate them. You argue that you didn't eat them; prove it!

Solution:

To prove that you did not eat the cookies, assume the opposite: Suppose you did eat the cookies. This means:

  • You should have already known about the cookies that were there, because you ate them.
  • But you knew you hadn’t touched them at all.
  • Checking your food diary, social plans, and other supplementary evidence will show that you were not there when they were eaten.
  • Therefore, the assumption that you ate the cookies conflicts with the factual details you gathered.

This provides indirect evidence that another potential culprit, such as your roommate, may be responsible.

Limitations and considerations

Although indirect evidence is a powerful technique, it is important to acknowledge its limitations:

  • Indirect proofs rely heavily on finding an actual contradiction. If this is overlooked, the proof attempt fails.
  • Sometimes, it can unnecessarily complicate simple problems. Therefore, it should be avoided when simpler proof methods exist.

Advanced example: Sum of two irrational numbers

Let's explore a complex mathematical statement:

Statement: Prove that the sum of two irrational numbers can be rational.

Evidence:

Suppose, for contradiction, that the sum of two irrational numbers is always irrational.

  • Consider two specific irrational numbers, x = pi and y = -pi.
  • By assumption, x + y must be irrational. However, pi + (-pi) = 0, which is rational.

Thus, this assumption leads to a contradiction, proving that the sum of two irrational numbers can in fact be rational.

Conclusion

Indirect proof is a versatile and beautiful method in mathematical logic and reasoning. It leverages the technique of contradiction to demonstrate the truth of a statement. In various branches of mathematics, it often uncovers relationships that are not easily accessible by direct methods.

By incorporating assumptions, deducing logical consequences, and clarifying contradictions, learners can master the application of indirect proofs in a variety of mathematical contexts. Through practice and further exploration, its beautiful utility in dissecting complex problems emerges in its full potential.


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Indirect Proof

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