Topic 1.1 Mathematical Statements

Question 1

What are mathematical statements generally made about?

Answer 1

Numbers and functions

Question 2

How do we represent mathematical statements?

Answer 2

In lowercase letters

Question 3

What must mathematical statements include?

Answer 3

A definition of all variables in the statement, because otherwise there is no way to ascertain whether the result will be true or false, which is the overall goal

Question 4

In terms of mathematical statements, are the terms ‘valid’ and ‘true’ synonymous?

Answer 4

No

Question 5

How do I ascertain if an ‘and’ statement is true?

Answer 5

Each component of the statement must be true for an ‘and’ statement to be true; if any are false, then the entire thing is false

Question 6

How do I ascertain if an ‘or’ statement is true?

Answer 6

At least one, or possibly both, of the statements must be true

Question 7

In an ‘or’ statement, one of the components MUST be false

Answer 7

False

Question 8

A letter can be a statement

Answer 8

True

Question 9

A number can be a statement

Answer 9

False, a number cannot be a statement as a number cannot be true or false

Question 10

What is a conjunction in mathematical statements?

Answer 10

A conjunction of two statements refers to the “and” statements; denoted as p∧q; resemblant of an intersection ∩

Question 11

When is the conjunction true?

Answer 11

The conjunction is true when both p and q are true

Question 12

What is a disjunction in mathematical statements?

Answer 12

A disjunction of two statements refers to the “or” statements; denoted as p∨q; resemblant of a union ∪

Question 13

When is the disjunction true?

Answer 13

The disjunction is true when at least one of p or q is true

Question 14

How else can you symbolise a negation ∽p?

Answer 14

⇁p

Question 15

What does ‘p if and only if q’ mean?

Answer 15

It means the p is positive only when q is negative

Question 16

When is the negation of p true?

Answer 16

When p is false

Question 17

Where does confusion relating to mathematical statements arise?

Answer 17

Confusion can arise from trying to interpret a true or false conclusion from mathematics that may not even make sense

Question 18

What does this symbol mean (∈)?

Answer 18

In

Question 19

What is another word for a condition?

Answer 19

Predicate

Question 20

Why must one discern between the phrasing ‘there exists’ and ‘for every’?

Answer 20

When being asked ‘for every’, you’re being asked if each element in the set complies with the condition, however, ‘there exists’ simply means that a single element exists that fits the condition. Subsequently, you must be careful, as ‘for every’ can even apply to empty sets.

Question 21

What is the name and meaning of the following symbol (∃)?

Answer 21

It is called the existential quantifier, and it means ‘there exists’

Question 22

What is the name and meaning of the following symbol (∀)?

Answer 22

It is called the universal quantifier, and it means ‘For all’