Proof is a series of logical steps , starting from given assumptions, that shows whether or not a result is true.

Define proof by deduction .

Proof by deduction uses a mathematical and logical argument to show whether or not a result is true.

What is the first thing you need to do when writing a proof by deduction?

Set up the general numbers you are working with, using "Let..." statements.

Why does reaching the form 2k prove that an expression is even?

Because even means the same as multiple of 2 , and 2k is 2 multiplied by an integer.

True or False? Every integer is also a rational number.

True. The integers sit inside the rationals, since any integer can be written as a fraction with denominator 1.

In a proof, why do you use a general expression such as 2n rather than particular even numbers?

A proof must hold for every even number. Working with 2n covers them all at once, whereas particular numbers only ever cover the cases you chose.

Define proof by exhaustion .

Proof by exhaustion shows that a result works for every allowed value , by testing all of them.

When can proof by exhaustion not be used?

When the allowed values cannot all be tested . You could not prove that every multiple of 4 is even this way, because there are infinitely many of them.

True or False? Checking that a result works for one or two values proves that it is true for all values.

False. A few examples are not a proof . Every allowed value has to be tested.

What must you show for each case in a proof by exhaustion?

The working for every case , not just the conclusion. Show the values you get and why each one fits the result being proved.

Define counter example .

A counter example is a value that does not work for a given statement, showing that the statement is not true in general.

True or False? Trigonometry is a good place to look for counter examples.

True. Several different angles can share the same sine value , so claims about angles often fail for a second angle you were not expecting.

True or False? Finding a counter example proves that a statement is false for every value.

False. It only shows the statement is not true in general . The statement may still hold for many other values.

When looking for a counter example, why should you check what type of number a statement is claimed for?

The claim may cover only integers (for example) rather than all real numbers. That changes which values you are allowed to use as a counter example.

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Proof (OCR A Level Maths A: Pure): Flashcards

Exam code: H240

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Cards in this collection (28)

Define proof.
Proof is a series of logical steps, starting from given assumptions, that shows whether or not a result is true.
What is the difference between $=$ and $\equiv$ ?
$=$ means the two sides are equal for some values.
$\equiv$ means the two sides are identical, so they are equal for all values.
In a proof, an even number is written as 2n, and an odd number is written as $______$ .
In a proof, an even number is written as 2n, and an odd number is written as 2n − 1.
You may also see an odd number written as 2n + 1.
What does $P \Rightarrow Q$ mean?
$P \Rightarrow Q$ means P implies Q: if P is true, then Q is true.
True or False?
$P \Leftrightarrow Q$ means that Q is true if and only if P is true.
True.
$P \Leftrightarrow Q$ means P and Q are equivalent, so each one being true forces the other to be true. This is stronger than $P \Rightarrow Q$ , which only works in one direction.
How do you write three consecutive integers algebraically?
Write them as $n - 1$ , $n$ , $n + 1$ .
Start in the middle, as it keeps the algebra simpler than starting at the lowest value.
A multiple of n can be written as $______$ , where k is an integer.
A multiple of n can be written as kn, where k is an integer.
Why is it useful to know that the square of any real number is $\geq 0$ ?
Multiplying or dividing an inequality by a positive number keeps the inequality sign the same way round. Since a square is never negative, a non-zero squared term is guaranteed to be positive.
Define proof by deduction.
Proof by deduction uses a mathematical and logical argument to show whether or not a result is true.
What is the first thing you need to do when writing a proof by deduction?
Set up the general numbers you are working with, using "Let..." statements.
Complete the working that shows the square of an even number is even: $(2 n)^{2} \equiv 4 n^{2} \equiv 2 (______)$
$(2 n)^{2} \equiv 4 n^{2} \equiv 2 (2 n^{2})$
This is 2 times an integer, so the result is even.
Why does reaching the form 2k prove that an expression is even?
Because even means the same as multiple of 2, and 2k is 2 multiplied by an integer.
True or False?
Every integer is also a rational number.
True.
The integers sit inside the rationals, since any integer can be written as a fraction with denominator 1.
In a proof, why do you use a general expression such as 2n rather than particular even numbers?
A proof must hold for every even number. Working with 2n covers them all at once, whereas particular numbers only ever cover the cases you chose.
What do the symbols $ℕ$ , $ℤ$ , $ℚ$ and $ℝ$ stand for?
$ℕ$ natural numbers, $ℤ$ integers, $ℚ$ rational numbers, $ℝ$ real numbers.
Each set is contained in the next.
Define proof by exhaustion.
Proof by exhaustion shows that a result works for every allowed value, by testing all of them.
When can proof by exhaustion not be used?
When the allowed values cannot all be tested. You could not prove that every multiple of 4 is even this way, because there are infinitely many of them.
True or False?
Checking that a result works for one or two values proves that it is true for all values.
False.
A few examples are not a proof. Every allowed value has to be tested.
To prove that a number is prime, you only need to test factors up to its $______$ .
To prove that a number is prime, you only need to test factors up to its square root.
What must you show for each case in a proof by exhaustion?
The working for every case, not just the conclusion. Show the values you get and why each one fits the result being proved.
In a proof by exhaustion, if a question restricts a claim to integers in $1 \leq n \leq 4$ , which values must you test?
Only $n = 1$ , $2$ , $3$ and $4$ . Those four cases exhaust every allowed value, even if the result happens to hold more widely.
Define counter example.
A counter example is a value that does not work for a given statement, showing that the statement is not true in general.
To disprove a statement, you only need to find $______$ value(s) that do not work.
To disprove a statement, you only need to find one value that does not work.
Which kinds of number are often worth trying when looking for a counter example?
Numbers that behave unusually: 0, 2 (the only even prime), and irrational numbers such as $π$ , $e$ and $\sqrt{2}$ .
True or False?
Trigonometry is a good place to look for counter examples.
True.
Several different angles can share the same sine value, so claims about angles often fail for a second angle you were not expecting.
Give a counter example to show that the sum of three distinct prime numbers is not always odd.
$5 + 11 + 2 = 18$ , which is even.
Every prime except 2 is odd, and odd + odd = even, so the third prime has to be 2.
True or False?
Finding a counter example proves that a statement is false for every value.
False.
It only shows the statement is not true in general. The statement may still hold for many other values.
When looking for a counter example, why should you check what type of number a statement is claimed for?
The claim may cover only integers (for example) rather than all real numbers. That changes which values you are allowed to use as a counter example.

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