conjectures and counterexamples

What is a conjecture?

• A statement that is always true without exception.

• A statement or proposition believed to be true based on observations but not yet proven.

• A proven theorem that has been accepted by mathematicians.

• A hypothesis that can be tested through experimentation.

In the sequence 1, -1, 2, -2, 3, what is the next number?

• -2

• -3

• 4

• 3

What is the importance of counterexamples in mathematics?

• They are used to prove theorems without exceptions.

• Counterexamples are crucial for testing the validity of conjectures and ensuring that statements are accurate.

• They help in simplifying complex mathematical problems.

• Counterexamples are only relevant in geometry.

What is a counterexample?

• An example that supports a conjecture or statement.

• An example that disproves a conjecture or statement, showing that it is not universally true.

• A type of mathematical proof that confirms a statement.

• A hypothetical scenario used in logical reasoning.

How many counterexamples are needed to disprove a conjecture?

• Two counterexamples are needed to disprove a conjecture.

• Only one counterexample is needed to disprove a conjecture.

• Three counterexamples are needed to disprove a conjecture.

• No counterexamples are needed to disprove a conjecture.

How can visual representations help in understanding conjectures?

• They can make conjectures more complex and difficult to understand.

• They can illustrate patterns and relationships, making it easier to identify conjectures and counterexamples.

• They are only useful for artistic purposes and do not aid in understanding.

• They can replace the need for logical reasoning in mathematics.

If all squares are rectangles, is a square a counterexample to the conjecture 'not all rectangles are squares'?

• Yes, a square is a counterexample; it disproves the conjecture.

• No, a square is not a counterexample; it supports the conjecture.

• Yes, a square is a counterexample; it is not a rectangle.

• No, a square is a counterexample; it contradicts the definition of rectangles.

What does it mean if a conjecture is true?

• It means that it holds for all cases or examples considered.

• It means that it is only true in some cases.

• It means that it has not been proven yet.

• It means that it is a widely accepted theory.

Provide an example of a conjecture related to even numbers.

• All even numbers are divisible by 2.

• All even numbers are greater than 10.

• All even numbers are prime numbers.

• All even numbers are odd numbers.

Can a conjecture be proven true for some cases but false for others?

• No, a conjecture must be true in all cases to be proven.

• Yes, a conjecture can be true for some cases and false for others, which is why counterexamples are important.

• A conjecture can only be proven true if it is universally applicable.

• Yes, but only if it is a mathematical theorem.