Foundations of Pure Mathematics: Lecture 1
Dr. Charles Eaton
September 27th, 2016
Dr. Charles Eaton Lecture 1 September 27th, 2016 1 / 13 Section 1: The language of mathematics
In this section: • Reading, understanding and writing mathematics; • Brief introduction to sets; • Quantifiers; • Truth tables; • What constitutes a proof.
Dr. Charles Eaton Lecture 1 September 27th, 2016 2 / 13 1.1 Statements
Mathematics involves statements. We have to be fluent in understanding, writing and manipulating them, as they can often be very complex.
There are essentially two types of statement: predicates and propositions. A proposition generally consists of quantifiers and predicates.
Dr. Charles Eaton Lecture 1 September 27th, 2016 3 / 13 Definition A proposition is a sentence about specific mathematical objects which is either true or false (but not both).
A typical proposition is“For every integer n, n2 is even”.
A proposition need not be true. The point is that we might hope to prove or disprove it (although this might not be possible in practice).
The proposition above could have been written in any number of ways, e.g., “The square of every integer is even”, or “If n is an integer, then n2 is even”, or “Let n be an integer. Then n2 is even”.
Dr. Charles Eaton Lecture 1 September 27th, 2016 4 / 13 A proposition consists of quantifiers and predicates. In the example “For every integer n, n2 is even”: • “For every integer n” is a quantifier. It tells us what n is. • “n2 is even” is a predicate. It is a statement which is true or false depending on what n is. Note that the division into quantifier and predicate might depend on how the proposition is written.
Definition A predicate is a statement in free variables which becomes a proposition when values are assigned to the variables. A quantifier assigns values to variables (in one of several ways).
Dr. Charles Eaton Lecture 1 September 27th, 2016 5 / 13 Examples (i) “π < 4”is a proposition, as it could be true or false (it just happens to be true, but that’s not what’s important here. “π > 4” is also a proposition (it just happens to be false).
(ii) “n < 4”is not a proposition. It can’t be judged to be true or false since we don’t know what n is. If n = 3, then the statement “n < 4” is true, but it is false if n = 5. n < 4 is a predicate.
We could form propositions from this predicate by giving quantifiers. Three examples of how this could happen: • “If n = π, then n < 4” is a proposition (note that this is the same as “π < 4”). Here the quantifier is “n = π”. • “for all integers n, n < 4” is also a proposition. Here “for all integers n” is the quantifier. • “there exists an integer n such that n < 4. Here “there exists an integer n” is the quantifier.
Dr. Charles Eaton Lecture 1 September 27th, 2016 6 / 13 (iii) “1+2” is neither a proposition nor a predicate. It is meaningless to say it is true or false.
(iv) “m < n” is a predicate. “If m = 3 and n = 1, then m < n” is a proposition (it happens to be false). “for all integers m, there exists an integer n such that m < n” is a proposition. Quantifier is “for all integers m, there exists an integer n”. (v) “Every even integer greater than 2 may be written as the sum of two prime numbers” is a proposition. It’s called Goldbach’s Conjecture, and as yet it is unknown whether the proposition is true or not.
Dr. Charles Eaton Lecture 1 September 27th, 2016 7 / 13 1.2 Sets
A set is a collection of objects. If for example the objects are a, b and c, then write {a, b, c} for the set. We often name sets, e.g., S = {a, b, c}. The objects comprising a set are called its elements, or members. If S is a set, then write a ∈ S to mean that a is an element of S. The ‘defining’ characteristic of a set S is that x ∈ S is a predicate, i.e., is either true or false, but not both. Elements of sets are often numbers, but they can be anything, e.g., points in space, types of fruit or even other sets. The objects in a set do not have to be related,e.g., the following is a set: {3, banana, {1, 3}, a}. This makes the idea of a set (almost) as abstract as possible, and allows it to be applied everywhere in mathematics. We have already come across some sets: N, Z, Q and R. We can now write “let n ∈ N” instead of “let n be a natural number”. Dr. Charles Eaton Lecture 1 September 27th, 2016 8 / 13 The ‘definition’ of a set above is vague. Set theory is much more subtle than it looks, and the rigorous definition of a set is in the realm of logic. As far as most mathematicians are concerned (and that includes us), we can be content with “naive set theory” where we muddle along with the vague definition above. Russell’s paradox highlights the dangers (not examinable!).
See board for Russell’s paradox
Dr. Charles Eaton Lecture 1 September 27th, 2016 9 / 13 1.3 Quantifiers
Some quantifiers are used so frequently we use special symbols as a shorthand.
∃ - “there exists”: Let A be a set, and let p be a predicate such that for each x ∈ A the statement p(x) is a proposition. ∃ x ∈ A, p(x) means “there exists an element x of A such that p(x) is true”. Examples: 2 (i) ∃ x ∈ R, x + x − 2 = 0 means “there exists a real number x such that x2 + x − 2 = 0” (this happens to be true, but that is not relevant here). 2 (ii) ∃ x ∈ R, x < 0.
Dr. Charles Eaton Lecture 1 September 27th, 2016 10 / 13 ∀ - “for all”: Let A be a set, and let p be a predicate such that for each x ∈ A the statement p(x) is a proposition. ∀ x ∈ A, p(x) means “for all elements x of A, p(x) is true”. Examples: 2 2 (i) ∀ y ∈ R, y ≥ 0 means “every real number y satisfies y ≥ 0”. (ii) ∀ n ∈ Z, 2n + 1 is odd. 2 (iii) ∀ x ∈ R, x > x.
Dr. Charles Eaton Lecture 1 September 27th, 2016 11 / 13 Compound quantifiers Statements often involve several quantifiers, e.g.,
∀ x ∈ N, ∃ y ∈ N, y > x.
Let A and B be sets, and let p be a predicate such that for each x ∈ A and y ∈ B the statement p(x, y) is a proposition. (i) ∀ x ∈ A, ∃ y ∈ B, p(x, y) means “for all x ∈ A there exists y ∈ B such that p(x, y) is true” (ii) ∃ x ∈ A, ∀ y ∈ B, p(x, y) means “there exists x ∈ A such that for all y ∈ B, p(x, y) is true”) (iii) ∀ x ∈ A, ∀ y ∈ B, p(x, y) means “for all x ∈ A and y ∈ B, p(x, y) is true”. (iv) ∃ x ∈ A, ∃ y ∈ B, p(x, y) means “there exist x ∈ A and y ∈ B such that p(x, y) is true”.
Dr. Charles Eaton Lecture 1 September 27th, 2016 12 / 13 The order in which the quantifiers are written is very important. Consider:
(a) ∀ x ∈ N, ∃ y ∈ N, y > x (b) ∃ y ∈ N, ∀ x ∈ N, y > x Proposition (a) is clearly true, whilst (b) is false.
Examples on board
Dr. Charles Eaton Lecture 1 September 27th, 2016 13 / 13