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Direct Proof and Indirect proof: Meaning and examples

The document discusses mathematical proofs, detailing concepts such as theorems, propositions, axioms, and logical consistency. It explains proof methods including direct proof, which deduces conclusions from hypotheses, and indirect proof, which uses contradiction to establish the truth of a statement. Various examples illustrate the process of proof in both paragraph and two-column formats.

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Direct and Indirect proofs Dr Shaymaa Darwish
Structure of mathematic s Undefined terms Propositio ns and Theorems Definitions Axioms Nature of Mathematics • Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.
• Theorem : A statement that has been proven to be true. • Proposition : A declarative sentence that is true or false but not both Definitions Example: Line segment is part of line (True) 9=10 (False) • Axiom: A basic assumption about a mathematical situation. (a statement we assume to be true based on logic ). Example: Line can be extended infinitely 0 is a natural number
What is the proof? • A proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true  Starting points are the axioms “rules” accepted by everyone.  Previously established theorems may be used to deduce the new ones  Mathematical proof is absolute, which means that once a theorem is proved, it is proved for ever. Until proven though, the statement is never accepted as a true one. Remarks
There are many techniques that can be used to prove the statements. Proof Methods: There is no one ideal proof - a theorem can be established using different techniques and none of them will be better or worse (as long as they are all valid Types of proof Direct proof Indirect proof
Direct proof assumes a given hypothesis, or any other known statement, and then logically deduces a conclusion. Direct Proof : 1. State the given hypothesis 2. State what you need to prove 3. Draw a figure to guide you 4. Present the proof using preferred method (Paragraph or Two Column Methods) Hypothesi s considere d as True Start from assumptio n Come on the conclusio n
Paragraph Proof : The proof consists of a detailed paragraph explaining the proof process. The paragraph contains steps and supporting justifications which prove the statement true. The paragraph can be quite lengthy. When using this method, it can be easy to overlook critical steps and/or supporting reasons if you are not careful. Be sure to list your steps in chronological order, and support each step with a definition, theorem postulate and/or property.
Example 1.1… Given : M is the midpoint of AB Prove: Proof in Paragraph form: Given that the point M is the midpoint of the line segment AB, by the definition of midpoint, the measurements of AM and BM are equal. Since AM=BM then by definition of congruent segments, we can say that the line segment AM is congruent to line segment BM. So
Example 1.2… Given: is the midpoint of and . Prove: It is given that C is the midpoint of both and . Since C is the midpoint, we now that and because the midpoint of the segment divides the segment into two congruent segments. Knowing vertical angles are congruent, we have We can now state that by SAS because if two sides and included angle of one triangle are congruent to two sides and included angle of another triangle, the two triangles are congruent
Two Column Proof : The proof consists of two columns, The first column contains a numbered chronological list of steps, called Statements, leading to the desired conclusion. The second column contains the justifications, called Reasons, to support each step in the proof. Remember that justifications are definitions, postulates, theorems and/or properties. This format clearly displays each step in your argument and keeps your ideas organized.
Example 1.2… Given: is the midpoint of and . Prove: Statements Reasons 1. is the midpoint of and . Given 2. and the midpoint of the segment divides the segment into two congruent segments 3. vertical angles are congruent 4. SAS: if two sides and included angle of one triangle are congruent to two sides and included angle of another triangle, the two triangles
Prove that if n and m are both even, then n + m is even Excersice 1.1… Theorem: Sum of all four angles of a quadrilateral is 360°. Excersice 1.2…
Indirect Proof ( Proof by contradiction ): Assumes the hypothesis (if given) together with a negation of a conclusion to reach the contradictory statement. 1. Accept the given statement is true. 2. Assume the opposite of the statement to be proved. 3. State the reasons directly until there is a contradiction of the given or the other statements. 4. State that the assumption of the opposite of the statement to be proved must be false. 5. Follow Steps 3 and 4 of writing direct proofs. Assume to the contrary that conclusion is false Prove that our assumption was wrong Prove that conclusion is true
Example 1.3… Given : M is not the midpoint of AB Prove: Proof in Two Column form:
Example 1.4… Proof in Two Column form: If ABC is isosceles, then the measure of the base angles cannot be 92 . Prove this indirectly. ∘ Remember, to start assume the opposite of the conclusion. Statement Reason 1. ABC is isosceles Given 2. The measure of the base angles are 92∘ Assumption 3. If the base angles are 92 , then they ∘ add up to 184 . ∘ Addition of the two base angles 4. the three angle measures of the triangle ABC more than 180∘ Addition of the three angles 4. The base angles cannot be 92∘ contradicts the Triangle Sum Theorem that says the three angle measures of all triangles add up to 180∘
Prove that Excersice 1.3… Excersice 1.4… If ∠ A and ∠B are complementary then ∠ A 90 . Prove ≤ ∘ this by contradiction.
• A counterexample is just one example showing the statement to be false. While there may be many examples to choose from, all you need is one. Example 1.5 : For any positive integer m, if m is prime, then is also prime Counterexample: Assume m=2.
Direct Proof and Indirect proof: Meaning and examples