Introduction
Corollary is a term used to refer to a logical consequence of a previously established statement or theorem. In other words, it is an immediate and direct conclusion that follows from a pre-existing argument or proposition. The word corollary is derived from the Latin term “corollarium,” which means “a deduction or consequence.”
In mathematics, corollaries are often used to establish a new theorem or proposition that follows directly from a previously proven statement. For example, if we have already established the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can use it to derive several corollaries. One such corollary could be that in a right-angled triangle, the hypotenuse is always the longest side. Another corollary could be that the sum of the squares of the other two sides is always less than the square of the hypotenuse.
In geometry, corollaries are often used to provide additional information about a given figure. For example, if we have a circle and a tangent line that intersects the circle at a point, we can use the corollary of the tangent-chord theorem to establish that the angle between the tangent line and the chord is equal to the angle that the chord subtends on the opposite side of the circle.
In logic and philosophy, corollaries are used to establish new propositions that follow logically from a given set of premises. For example, if we assume that all men are mortal and that Socrates is a man, then we can establish the corollary that Socrates is mortal. In this case, the corollary follows directly from the premises and does not require any additional reasoning.
In science, corollaries are often used to make predictions or establish new relationships between different variables. For example, if we know that the boiling point of water decreases with increasing altitude, we can use this corollary to predict the boiling point of water at a given altitude. Similarly, if we know that a certain chemical reaction produces a certain amount of energy, we can use this corollary to establish the relationship between the amount of reactants and the amount of energy produced.
In economics, corollaries are often used to establish new relationships between different economic variables. For example, if we know that the demand for a certain product increases with decreasing price, we can use this corollary to establish the relationship between the price of the product and the quantity demanded.
In politics and social sciences, corollaries are often used to establish new relationships between different social or political variables. For example, if we know that the level of education of a population is positively correlated with the level of economic development, we can use this corollary to establish the relationship between education and economic development.
In summary, corollary is a term used to refer to a logical consequence of a previously established statement or theorem. It is often used in mathematics, geometry, logic, philosophy, science, economics, politics, and social sciences to establish new relationships, make predictions, or provide additional information. Corollaries are essential in building logical arguments and establishing new knowledge.
Definition of Corollary
A corollary is a statement that follows logically from a previously proven theorem or proposition. In other words, if a theorem or proposition has been proven to be true, then any statement that can be deduced from it is also true. A corollary is a direct consequence of the theorem or proposition and is often used to help prove other theorems or propositions.
Examples of Corollary
- Corollary of the Pythagorean Theorem
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. A corollary of the Pythagorean Theorem is that in any right-angled triangle, the length of the hypotenuse is always greater than the length of either of the other two sides.
- Corollary of the Triangle Inequality
The Triangle Inequality states that the sum of any two sides of a triangle is always greater than the third side. A corollary of the Triangle Inequality is that if the sum of the lengths of any two sides of a triangle is equal to the length of the third side, then the triangle must be degenerate, which means it is not a triangle at all.
- Corollary of the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus states that if a function f(x) is continuous on an interval [a,b], then the integral of f(x) from a to b can be found by evaluating F(b) – F(a), where F(x) is any antiderivative of f(x). A corollary of the Fundamental Theorem of Calculus is that if the function f(x) is differentiable on the interval [a,b], then the integral of f(x) from a to b is equal to the area under the curve of f(x) from a to b.
- Corollary of the Law of Sines
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. A corollary of the Law of Sines is that in any triangle, the product of the lengths of two sides divided by the sine of the included angle is equal to twice the area of the triangle.
- Corollary of the Gauss-Bonnet Theorem
The Gauss-Bonnet Theorem is a fundamental theorem in differential geometry that relates the curvature of a surface to its topology. A corollary of the Gauss-Bonnet Theorem is that the total curvature of a closed surface is equal to 2? times the Euler characteristic of the surface, where the Euler characteristic is a topological invariant that can be calculated from the number of vertices, edges, and faces of a polyhedron.
Conclusion
In conclusion, a corollary is an important concept in mathematics and other disciplines. It is a statement that follows logically from a previously proven theorem or proposition and is often used to simplify proofs and to build upon previously established results. The examples of corollaries discussed in this article highlight the significance of this concept in various areas of mathematics, including geometry, calculus, and topology.
Quiz
Are you ready to put your knowledge of corollaries to the test? Corollaries are fascinating logical statements that follow as a result of a previously proven theorem. In this 10-question quiz, you’ll have the opportunity to show off your understanding of these powerful mathematical tools. Let’s get started!
- What is a corollary in mathematics?
- How is a corollary different from a theorem?
- Can a corollary be proven on its own, without relying on a theorem?
- If a theorem is true, is its corollary also necessarily true?
- What is an example of a corollary in geometry?
- Can a corollary ever be more important than the theorem from which it follows?
- How are corollaries used in real-world applications of mathematics?
- Is it possible for a corollary to be used to disprove a theorem?
- Are corollaries exclusive to mathematics, or can they be found in other fields of study as well?
- Can corollaries be used to create new mathematical theories, or are they only used to explain previously proven theorems?
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