Proof by Deduction

Proof by Deduction

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The Concept of Proof by Deduction and its Fundamental Elements

Proof by deduction is a mathematical technique that involves proving the validity of a statement by relying on the truth of each individual component and the logical connections between them. Let's delve into the key components of this proof method.

Statement A

The first element of a proof by deduction is statement A, which serves as a premise and provides potential solutions to the statement being proven. For example, in the statement "If today is a weekend, then tomorrow must be a weekday," statement A can be Saturday or Sunday, as these are the only two days that constitute the weekend.

Statement B

The second element of a proof by deduction is statement B, which is also a premise and is combined with statement A to test the logical validity of the main statement. Continuing with our example, statement B would be "tomorrow is a weekday."

Concluding Statement

The final component of a proof by deduction relies on the truth of statement A and B. It is important to note that the strength of the logic of the statement is determined by the weakest premise. If either statement A or B is false, the concluding statement will also be false. This proof method heavily depends on mathematical axioms to support the logic of the main statement.

When working with mathematical axioms, it is essential to be familiar with algebraic rules to accurately express them in mathematical form. While these rules may seem familiar, it requires creativity to use them effectively. For instance, when representing a mathematical axiom as a formula, the variable "n" can be used to represent any number. Let's explore some examples to see how this is done.

  • Express n as a multiple of 12: To achieve this, simply multiply 12 by n, resulting in 12n.
  • Find the next two consecutive numbers after n: To do this, add 1 to each consecutive number. Therefore, the first term would be n + 1, the second term would be n + 2, and so on.

Examples of Proof by Deduction

Let's consider a couple of examples to understand how proof by deduction works in practice.

Example 1:

Conjecture: The sum of two consecutive numbers is equivalent to the difference between the two consecutive numbers squared.

Step 1: Begin by representing both consecutive numbers algebraically as n and n+1.

Step 2: The sum of the two consecutive numbers is then given by n + n + 1, which simplifies to 2n + 1.

Step 3: To find the difference between the two consecutive numbers squared, we square each term, resulting in n^2 and (n+1)^2. Expanding and simplifying these terms gives us n^2 + 2n + 1 and n^2 + 2n + 1, which are equal.

Step 4: Therefore, the concluding statement would be: "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."

Example 2:

Conjecture: The answer to the equation 2x^2 + 8x + 4 is always positive.

Step 1: To complete the square, begin by halving the coefficient of x (8) and substituting it into our equation, resulting in 2x^2 + 4x + 4.

Step 2: Expanding this gives us a constant of 4 outside of the bracket, but we need +20 to match the original equation. Therefore, we add 16 to get 2x^2 + 4x + 20.

Step 3: Now we can solve this equation by factoring and applying the quadratic formula, resulting in two solutions: x = -5 and x = -1.

Step 4: The concluding statement for this example would be: "Regardless of the value of x, by squaring it and adding 4, the value of the equation 2x^2 + 8x + 4 will always be positive."

Key Takeaways of Proof by Deduction

  • Proof by deduction heavily relies on mathematical axioms and logic to prove or disprove a conjecture.
  • Algebraic expressions can be used to represent various axioms, such as consecutive even and odd numbers.
  • To use the deduction theorem, one must consider the logic of the conjecture, represent any axioms mathematically, solve through to test the logic, and conclude with a statement about the truth of the conjecture.

The Fundamentals of Deduction

Unlike other proof methods, there are no explicit rules to follow when using deduction. Instead, the truth of the second statement in a conjecture is based on the truth of the first statement.

Who Introduced Deductive Proof?

The Origins of Deductive Reasoning: From Ancient Times to Aristotle

Deductive reasoning, a method of logical thinking, has been a part of human thought since ancient times. However, it was the Greek philosopher Aristotle who formalized this intellectual process.

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