Proof by contradiction, where assuming the negation of a statement leads to an inconsistency, and counterexamples used to disprove claims, often with examples involving prime numbers and rational numbers.
A rational number is one that can be written in the form
Note that any fraction can be reduced to the point where the numerator and denominator share no common factors:
To prove a statement by contradiction, we essentially show that if the statement were not true, math would "break".
In practice, we assume the opposite of the statement, and and derive a contradiction - a mathematical inconsistency.
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, the set of prime numbers begins as
Notice that 1 is not included in this list because it has only one positive divisor (namely, itself), and thus does not meet the definition.
The easiest way to show a statement is not true is to find a specific example for which it is not true.
Nice work completing Contradiction and Counterexamples, here's a quick recap of what we covered:
Exercises checked off
Proof by contradiction, where assuming the negation of a statement leads to an inconsistency, and counterexamples used to disprove claims, often with examples involving prime numbers and rational numbers.
A rational number is one that can be written in the form
Note that any fraction can be reduced to the point where the numerator and denominator share no common factors:
To prove a statement by contradiction, we essentially show that if the statement were not true, math would "break".
In practice, we assume the opposite of the statement, and and derive a contradiction - a mathematical inconsistency.
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, the set of prime numbers begins as
Notice that 1 is not included in this list because it has only one positive divisor (namely, itself), and thus does not meet the definition.
The easiest way to show a statement is not true is to find a specific example for which it is not true.
Nice work completing Contradiction and Counterexamples, here's a quick recap of what we covered:
Exercises checked off