Topic 1.1 Mathematical Statements Flashcards
What are mathematical statements generally made about?
Numbers and functions
How do we represent mathematical statements?
In lowercase letters
What must mathematical statements include?
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
In terms of mathematical statements, are the terms ‘valid’ and ‘true’ synonymous?
No
How do I ascertain if an ‘and’ statement is true?
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
How do I ascertain if an ‘or’ statement is true?
At least one, or possibly both, of the statements must be true
In an ‘or’ statement, one of the components MUST be false
False
A letter can be a statement
True
A number can be a statement
False, a number cannot be a statement as a number cannot be true or false
What is a conjunction in mathematical statements?
A conjunction of two statements refers to the “and” statements; denoted as p∧q; resemblant of an intersection ∩
When is the conjunction true?
The conjunction is true when both p and q are true
What is a disjunction in mathematical statements?
A disjunction of two statements refers to the “or” statements; denoted as p∨q; resemblant of a union ∪
When is the disjunction true?
The disjunction is true when at least one of p or q is true
How else can you symbolise a negation ∽p?
⇁p
What does ‘p if and only if q’ mean?
It means the p is positive only when q is negative
When is the negation of p true?
When p is false
Where does confusion relating to mathematical statements arise?
Confusion can arise from trying to interpret a true or false conclusion from mathematics that may not even make sense
What does this symbol mean (∈)?
In
What is another word for a condition?
Predicate
Why must one discern between the phrasing ‘there exists’ and ‘for every’?
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.
What is the name and meaning of the following symbol (∃)?
It is called the existential quantifier, and it means ‘there exists’
What is the name and meaning of the following symbol (∀)?
It is called the universal quantifier, and it means ‘For all’
