MTH1020

Question 1

What is a proof?

Answer 1

Mathematical proofs establish the truth of a statement to an extremely high degree of certainty. Necessarily and logically true.

Question 2

Counterexample

Answer 2

An example which disproves a statement. Just one set of values that disproves a statement.

Question 3

□

Answer 3

QED - goes at the end of every proof

Question 4

How are conditional statements asked?

Answer 4

If….Then…
If P, then Q

Question 5

A conditional statement asserts that…

Answer 5

If one statement (P) is true, then another statement (Q) is true.

Question 6

In conditional statements P is…

Answer 6

The hypothesis or assumption

Question 7

In conditional statements Q is…

Answer 7

The conclusion

Question 8

If given a conditional statement, prove by:

Answer 8

Direct proof

Question 9

What is a direct proof?

Answer 9

Assume hypothesis is true, and then show why the conclusion logically follows.

Question 10

An odd number is…
If something is (odd/even), then something else is (odd/even) type of questions.

Answer 10

• an integer which is not divisible by 2
• i.e. a number of the form 2k + 1 for some integer k

Question 11

An even number is…
If something is (odd/even), then something else is (odd/even) type of questions.

Answer 11

• an integer divisible by 2
• i.e. a number of the form 2k for some integer k

Question 12

What is the negation of a statement?

Answer 12

The negation of a statement P is the assertion that P is false. Often abbreviated to ‘not P’

Question 13

Negating an ‘and’ statement leads to a…
(and vice versa)
e.g. a=0 and b=0

Answer 13

‘or’ statement
e.g. a≠0 or b≠0

Question 14

‘or’ statements in mathematics - inclusive.
“P or Q” means…

Answer 14

“P is true or Q is true or both”, or equivalently, “at least one of the statements P and Q is true”.

Question 15

‘exclusive or’ often written as “xor” in mathematics.
“P xor Q” means…

Answer 15

“P is true or Q is true but NOT both”, or equivalently, “exactly one of the statements P and Q is true”.

Question 16

De Morgan’s Laws:

Answer 16

Not (P and Q) is equivalent to (Not P) or (Not Q)

Not (P or Q) is equivalent to (Not P) and (Not Q)

Question 17

Proof by Contradiction

Answer 17

suppose the contrary

Question 18

Steps for doing a proof by contradiction, which proves a statement P as follows:

Answer 18

1. Assume statement P is false (assume the negation of P)
2. Under this assumption, show that a contradiction or nonsensical statement follows. (most important)
3. Conclude that it was wrong to assume that P is false
4. Conclude that P is true

Question 19

If x is positive and rational then…

Answer 19

Then x can be written as a fraction: x = m/n , where m and n are positive integers and n≠0

Question 20

Prove that something is NOT rational (irrational)

Answer 20

x=m/n
As m,n are both even with the highest common factor 1.

Question 21

What is the converse of the conditional statement P=>Q?

Answer 21

Q=>P
=> implies (Q implies P)

Question 22

The converse of the statement “If P then Q” is..

Answer 22

“If Q then P”

Question 23

What is the contrapositive of the conditional statement P=>Q?

Answer 23

The contrapositive of the statement P=>Q is the statement (Not Q) => (Not P)

Question 24

A statement is logically equivalent to its…

Answer 24

contrapositive

Question 25

If the original statement is true, the contrapositive is...

Answer 25

Also true

Question 26

If the original statement is false, the contrapositive is...

Answer 26

Also false

Question 27

Proving contrapositive example: If x is irrational, then 2x + 1 is irrational. What is the contrapositive? (If something, then something - the same e.g. in this case 'irrational')

Answer 27

Contrapositive: If 2x + 1 is rational, then x is rational.

Question 28

Set notation: N=?

Answer 28

Natural numbers {1, 2, 3, 4...}

Question 29

Set notation: Z=?

Answer 29

Integers {...-2, -1, 0, 1, 2, 3...}

Question 30

Set notation: Q=?

Answer 30

Rational numbers/fractions {a/b : a,b ∈ Z and b≠0}

Question 31

Set notation: R=?

Answer 31

Real numbers (number line)

Question 32

Set notation: C=?

Answer 32

Complex numbers (watch this space)

Question 33

Equivalent Statements - Two statements P and Q are (logically) equivalent if...

Answer 33

Both P => Q And Q => P are true

Question 34

P => Q and Q => P is often abbreviated to...

Answer 34

P <=> Q

Question 35

When P an Q are equivalent statements...

Answer 35

P is true precisely when Q is true

Question 36

For equivalent statements, P is true if and only if...

Answer 36

Q is true

Question 37

iff

Answer 37

if and only if

Question 38

To prove P and Q are equivalent you need to...

Answer 38

Prove 2 things: P => Q Q => P

Question 39

Divisibility definition: let m and n be integers

Answer 39

When there exists an integer k such that m=kn, we say that m is divisible by n. OR m is the multiple of n. OR n is a divisor or factor of m.

Question 40

Define a prime number

Answer 40

A positive integer whose only positive divisors are 1 and itself

Question 41

Define a composite

Answer 41

A positive integer which has a positive divisor other than 1 and itself

Question 42

An integer is even when...

Answer 42

it is divisible by 2

Question 43

"For all"

Answer 43

A (upside down) - change these to pictures?

Question 44

"There exists"

Answer 44

E (backwards)

Question 45

A (upside down)

Answer 45

"For all"

Question 46

E (backwards)

Answer 46

"There exists"

Question 47

A "for all" or universal statement claims that...

Answer 47

a certain property holds for all elements of the set

Question 48

A "there exists" or existence statement claims...

Answer 48

a certain property holds for at least one element of a set

Question 49

Statements involving "for all" or "there exists" are...

Answer 49

quantified - they claim a certain "quantified" number of elements of a set have a certain property.

Question 50

The phrases "for all" and "there exists" are..

Answer 50

quantifiers

Question 51

To prove a universal statement you need to explain...

Answer 51

why every element of a set has the claimed property.

Question 52

To disprove a universal statement...

Answer 52

you only need to find one counterexample