What is a Corollary?

S8-SA2-0012

Grade Level:

Class 7

AI/ML, Data Science, Research, Journalism, Law, any domain requiring critical thinking

Definition

What is it?

A corollary is a statement or result that logically follows directly from a previously proven statement or theorem, often with little or no additional proof needed. Think of it as a 'mini-conclusion' that comes naturally from a bigger truth we already know.

Simple Example

Quick Example

If we know that 'All students who score above 90% in exams get a scholarship', then a corollary would be 'Rohan, who scored 95%, will get a scholarship'. Rohan's scholarship is a direct and obvious consequence of the main rule.

Worked Example

Step-by-Step

Let's say we have a rule: 'If a cricket team scores more than 200 runs in a T20 match, they usually win.'
---STEP 1: Understand the main statement. The main statement is that scoring over 200 runs leads to a win.
---STEP 2: Observe a specific situation. In today's match, Team India scored 210 runs.
---STEP 3: Apply the main statement to the situation. Since 210 is more than 200, Team India met the condition.
---STEP 4: State the direct conclusion. Therefore, a corollary is that 'Team India will likely win today's match.' This outcome directly follows from our initial rule.

Why It Matters

Understanding corollaries helps you think logically and make quick, accurate deductions in many fields. In Data Science, if a model predicts a trend, a corollary might be a specific business action. Researchers use them to quickly derive new insights from established theories.

Common Mistakes

MISTAKE: Confusing a corollary with a completely new idea or a separate theorem. | CORRECTION: A corollary is always a direct and obvious consequence of an already established truth, not a brand new discovery requiring extensive proof.

MISTAKE: Believing a corollary needs a lot of extra proof. | CORRECTION: Corollaries need very little, if any, additional proof because their truth is self-evident once the main statement is accepted.

MISTAKE: Thinking a corollary is just any related statement. | CORRECTION: It must be a *direct and immediate* logical consequence. If it requires many steps or new assumptions, it's probably not a corollary.

Practice Questions

Try It Yourself

QUESTION: If we know 'All squares are rectangles', what is a corollary about a shape named 'Boxy' which is a square? | ANSWER: Boxy is a rectangle.

QUESTION: Main statement: 'If a person studies diligently every day, they will likely score good marks in their exams.' What is a corollary for Priya, who studies 3 hours every day? | ANSWER: Priya will likely score good marks in her exams.

QUESTION: Theorem: 'The sum of angles in any triangle is 180 degrees.' What is a corollary if you have a right-angled triangle (one angle is 90 degrees)? | ANSWER: The sum of the other two angles in a right-angled triangle must be 90 degrees (180 - 90 = 90).

MCQ

Quick Quiz

Which of the following best describes a corollary?

A completely new scientific discovery.

A statement that contradicts a main theorem.

A direct and obvious consequence of an already proven statement.

A complex proof requiring many new assumptions.

The Correct Answer Is:

A corollary is defined as a direct and obvious consequence of a previously proven statement, requiring little to no additional proof. Options A, B, and D describe things that are not corollaries.

Real World Connection

In the Real World

In law, if a judge rules that 'all contracts signed by minors are invalid', a corollary would be that a specific contract signed by a 15-year-old is invalid. This allows lawyers to quickly apply established rulings to new cases without re-proving the main point. Similarly, in journalism, if a news report confirms a major policy change, a corollary could be its immediate impact on daily life for citizens.

Key Vocabulary

Key Terms

THEOREM: A statement that has been proven true | DEDUCTION: The process of reaching a conclusion by reasoning | LOGICAL: Following the rules of logic or clear reasoning | CONSEQUENCE: A result or effect of an action or condition

What's Next

What to Learn Next

Now that you understand corollaries, you can explore 'Theorems and Proofs'. Learning about proofs will help you understand how those main statements are established, from which corollaries then logically flow. Keep building your critical thinking skills!