Proper notes

Question 1

Statements (2)

Answer 1

• Can be either true or false but not both
• Can be combined to make “bigger statements”

Question 2

Same/logically equivalent

Answer 2

Two statements are the same – logically equivalent – when they have identical truth tables

Question 3

not A

Answer 3

• Turns false statements into true statements and vice versa
• Applies only to what immediately follows it unless brackets are used
• not not A is logically equivalent to A

Question 4

A and B

Answer 4

• True only when both A and B are true, otherwise false
• Symmetry: A and B is logically equivalent to B and A
• (A and B) and C is logically equivalent to A and (B and C) and is also logically equivalent to A and B and C

Question 5

A or B

Answer 5

• True when either A or B or both are true – i.e., true when at least one of the two statements A, B, is true
• Symmetry: A or B is logically equivalent to B or A
• (A or B) or C is logically equivalent to A or (B or C) and is also logically
  equivalent to A or B or C

Question 6

Negating compound statements

Answer 6

• not (A and B) is logically equivalent to not A or not B
• not (A or B) is logically equivalent to not A and not B

Question 7

if A then B

Answer 7

• Also written as B if A or as A only if B
• Not symmetric: if A then B is not the same as if B then A

Question 8

A if and only if B

Answer 8

• Also written as A iff B
• Symmetric: A iff B is logically equivalent to B iff A
• Equivalent to (if B then A) and (if A then B)
• Equivalent to (A if B) and (A only if B)

Question 9

give converse

Answer 9

Question 10

Converse of logically equivalent statements

Answer 10

Question 11

Give contrapositives

Answer 11

Question 12

Relationship between a statement and its contrapositive

Answer 12

They are logically equivalent

Question 13

Give contrapositive

Answer 13

Question 14

Give contrapositive

Consider the statement (about integers):
if a and b are odd, then ab is odd.

Answer 14

Question 15

If you are asked for a sufficient/necessary condition for B to be true then you need to look for ….

Answer 15

• If you are asked for a sufficient condition for B to be true then you need to look for a condition that guarantees to make B true. (but not necessarily the other way round)
• If you are asked for a necessary condition for B to be true then you need to look for something that must be the case for B to be true but might not be enough by itself to guarantee that B is true.

Question 16

Fill in how we said before, and by using symbols (implies)

Answer 16

Question 17

• 2 important notes about ‘for all’

Answer 17

1. Mathematicians like to say what their statements apply to and sometimes they do this using phrases like “for all”.
2. Often a statement can be true only in certain situations and mathematicians can use phrases like “for all….” to make it clear what circumstances they are considering.

Question 18

• 2 other phrases which have same meaning as ‘for all’

Answer 18

Question 19

‘There exists’ meaning

Answer 19

• for some (meaning for at least one)
• Mathematicians also like to assert that something [some mathematical thing, like a number or a function etc.] can be found to make something true, in these cases they tend to use the term “there exists” [usually along with the phrase ‘such that’]

Question 20

2 ways of saying ‘there exists’

Answer 20

Question 21

Thinking informally about ‘for all’ and ‘there exists’

Answer 21

• When you see the phrase for all x … you can think of it as telling you that you can pick ANY x you want from the given set of xs and then the corresponding statement will be true. The phrase is telling you that every value of x makes the statement true.
• And when vou see the phrase there exists an x such that… you can think of it as issuing a challenge: you are challenged to FIND an x that makes the statement that the phrase is applied to true. The phrase is telling you that there is at least one x that makes the statement true.

Question 22

What does there exists say about any values for which the corresponding statement is false.

Answer 22

Be aware that there exists does not mean that there are any values for which the corresponding statement is false

Question 23

See page 52, combining the two phrases together

Answer 23

Ask mr barker for help

Question 24

3 steps in negating this statement

Answer 24

Question 25

Summary of Negating 'for all' and 'there exists'

Answer 25

Question 26

NOT necessary, but may be impressive to know - what do mathematicians call the phrases 'for all' and 'there exists' ?

Answer 26

In addition, it is worth noting that mathematicians call the phrases for all and there exists quantifiers: for all is known as the universal quantifier (because it sets the universe of things that you are allowed to consider), and there exists is known as the existential quantifier.

Question 27

Name 4 methods of proof

Answer 27

- Simple deductive proofs - Proof by contradiction - Proof by contrapositive - Disproof by counterexample

Question 28

Prove

Answer 28

Question 29

General structure of simple deductive proofs

Answer 29

Question 30

Prove

Answer 30

Question 31

General structure of proof by contradiction

Answer 31

Question 32

Why does proof by contrapositive work

Answer 32

* If we are asked to prove **if A then B** we can try to prove the contrapositive instead as sometimes this can turn out to be much easier. * Remember that the contrapositive of **if A then B** is **if not B then not A** and these statements are logically equivalent – i.e. both expressions say the same thing. * Because **if not B then not A** is the very same thing as **if A then B** we can prove the contrapositive of a statement instead of proving the statement itself.

Question 33

Prove

Answer 33

Question 34

Simple counterexample proof

Answer 34

Question 35

How might we set about finding a counterexample to a statement of the form **if A then B**?

Answer 35

* First we need to keep in mind that a counterexample is an example where the statement [in this case our statement is **if A then B**] is false, * so we need to find an example for **A** and for **B** such that **if A then B** is false: the only way that **if A then B** can be false is if we can find an example of statement **A** that is true and an example of statement **B** that is false.

Question 36

Prove

Answer 36

Question 37

2 common mathematical errors in purported proofs

Answer 37

- claiming 'if ab = ac, then b = c' - assuming 'if sin A = sin B, then A = B' neither of these are valid deductions

Question 38

We need to be careful when squaring and square rooting equations

Answer 38

We need to be careful in case we generate extra solutions to the equation

Question 39

3 pitfalls that you need to watch out for when dealing with inequality signs

Answer 39

Question 40

Can you spot exactly where the error occurs?

Answer 40

- Dividing both sides of an expression by a second expression that is equal to zero can cause problems. - Generally, we cannot divide by zero as it can generate nonsense. - For instance, we know 7 × 0 = 5 × 0 but we cannot divide both sides by 0 to give 7 = 5. - This issue extends to examples that contain algebra. Here is a classic proof that commits this error

Question 41

Slightly hand-wavy r2drew2 list of statements and their negation

Answer 41

Note last statement has 2 possible negations which are both equivalent. (Also it should really say All A are not B)

Question 42

Which statement is true, and which is false and why

Answer 42

Question 43

What do we take from this example? (2)

Answer 43

* What we take from these two examples is that the order of the phrases **for all** and **there exists** is important when they occur together and we have to respect the order in which they appear. * Only once we have dealt with the first phrase can we then deal with the second phrase in light of what the first phrase has told us.

Question 44

A final note on the order of phrasing

Answer 44