What is proof in mathematics?

Proof in mathematics is the use of a series of logical steps to show that a mathematical result is true for all specified cases. E.g. showing that the square of an odd number is odd for every odd number.

True or False? Seeing that a result works for a few numbers is enough to show it will work for all numbers .

False. Seeing that a result works for a few numbers is not enough to show it will work for all numbers . A valid proof is required to show that it works for all values.

What is proof by deduction ?

A proof by deduction uses known facts, and logical steps and arguments connecting those facts, to show that a mathematical result is true.

What is proof by exhaustion ?

Proof by exhaustion is a way of proving a result by showing that the desired result works for every allowed value . This is a good method when there are only a limited number of cases or allowed values to check.

What is disproof by counter-example ?

Disproof by counter-example is a way of proving that a result is not true , by finding a case or value that does not work in the result. That case or value is called a counter - example

True or False? To disprove a result by counter-example , you need to show that the result does not work for every possible case or value.

False. To disprove a result by counter-example , you only need to show that the result does not work for one particular case or value.

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What is proof in mathematics?

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What is proof in mathematics?
Proof in mathematics is the use of a series of logical steps to show that a mathematical result is true for all specified cases.
E.g. showing that the square of an odd number is odd for every odd number.
True or False?
Seeing that a result works for a few numbers is enough to show it will work for all numbers.
False.
Seeing that a result works for a few numbers is not enough to show it will work for all numbers.
A valid proof is required to show that it works for all values.
What symbol is used to denote the set of all integers?
The set of all integers is denoted by the symbol $ℤ$ .
What symbol is used to denote the set of all positive integers?
The set of all positive integers (1, 2, 3, ...) is denoted by the symbol $ℤ^{+}$ .
What is $ℕ$ the symbol for?
The symbol $ℕ$ denotes the set of all natural numbers (0, 1, 2, 3, ...).
What is $ℚ$ the symbol for?
$ℚ$ is the symbol for the set of all rational numbers.
I.e., the set of all numbers that can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
Define the term mathematical identity.
A mathematical identity is a statement that is true for all possible values of any variables used (for example $x$ or $θ$ ).
What symbol is used to denote a mathematical identity?
A mathematical identity is denoted by the symbol $\equiv$ .
What is proof by deduction?
A proof by deduction uses known facts, and logical steps and arguments connecting those facts, to show that a mathematical result is true.
What algebraic convention is used to represent even numbers in proofs?
In proofs, even numbers are typically represented as $2 n$ , where $n$ is an integer.
( $2 k$ or $2 m$ are also often used in this way.)
What algebraic convention is used to represent odd numbers in proofs?
In proofs, odd numbers are typically represented as $2 n - 1$ , where $n$ is an integer.
$2 n + 1$ may also be used.
(Variables $k$ or $m$ are also often used instead of $n$ in this context.)
What is proof by exhaustion?
Proof by exhaustion is a way of proving a result by showing that the desired result works for every allowed value.
This is a good method when there are only a limited number of cases or allowed values to check.
What is disproof by counter-example?
Disproof by counter-example is a way of proving that a result is not true, by finding a case or value that does not work in the result.
That case or value is called a counter-example
True or False?
To disprove a result by counter-example, you need to show that the result does not work for every possible case or value.
False.
To disprove a result by counter-example, you only need to show that the result does not work for one particular case or value.

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