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Contradictories are a fundamental concept in mathematics, particularly in the field of logic. They are pairs of statements where one statement negates the other, meaning that if one statement is true, the other must be false, and vice versa. This concept is crucial in mathematical proofs and logical reasoning.

1. Definition: Contradictories are defined as a pair of propositions or statements where one is the negation of the other. For instance, if we have a statement P, the contradictory of P will be not P, often represented as ~P.

2. Properties: The main property of contradictories is that they cannot both be true and they cannot both be false at the same time. If P is true, then ~P must be false. If P is false, then ~P must be true. This is known as the principle of contradiction.

3. Use in Mathematical Proofs: Contradictories are often used in mathematical proofs, particularly in proof by contradiction (also known as reductio ad absurdum). In this method, we assume that the statement we want to prove is false, and then show that this assumption leads to a contradiction, thus proving that the original statement must be true.

4. Examples: An example of contradictories could be the statements “The number is even” (P) and “The number is not even” (~P). If a given number is even (P is true), then it cannot be not even (~P is false). Conversely, if a number is not even (~P is true), then it cannot be even (P is false).

5. Contradictories vs. Contraries: It’s important to distinguish contradictories from contraries. Contraries are pairs of statements that cannot both be true at the same time, but they can both be false. For example, “The number is even” and “The number is odd” are contraries, not contradictories. A number cannot be both even and odd at the same time, but it can be neither (in the case of non-integers).

In conclusion, the concept of contradictories is a cornerstone of mathematical logic and reasoning. Understanding this concept can greatly aid in the comprehension and construction of mathematical proofs.

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