Unit 1: Number systems and sets

• Every natural number is a whole number, every whole number an integer, every integer a rational number, and rational and irrational numbers together form the real numbers.

• Converting a repeating decimal: let x = 0.272727…; then 100x = 27.2727…; subtracting, 99x = 27, so x = 27/99 = 3/11.

• Properties: rational numbers are closed under addition, subtraction, multiplication and division (by non-zero numbers); addition and multiplication are commutative and associative; 0 is the additive identity and 1 the multiplicative identity; between any two rational numbers there is another (their average).
• The sum or product of a rational and an irrational number (non-zero rational for products) is irrational — e.g., 2 + √3 and 3√2.

• Computing link: 2¹⁰ = 1,024 bytes = 1 KB; 2²⁰ = 1 MB; scientific notation writes 0.000045 as 4.5 × 10⁻⁵.

• Radical: ⁿ√a is the number whose nth power is a; √a is the square root and ∛a the cube root. In exponent form, ⁿ√a = a^(1/n).

• Rationalising the denominator: 1/√3 = √3/3; 1/(√5 − √2) = (√5 + √2) ÷ (5 − 2) = (√5 + √2)/3.

A set is a well-defined collection of distinct objects. "Well-defined" means that for any object we can say clearly whether it belongs to the set or not. "The vowels of English" is a set; "the good students of a class" is not, because "good" is a matter of opinion. The objects in a set are called its elements or members. We write sets with capital letters (A, B, C) and elements with small letters. If a is an element of A, we write a ∈ A; if not, a ∉ A.

• Order does not matter: {1, 2, 3} and {3, 1, 2} are the same set.
• Repetition does not matter: {1, 1, 2} is the same as {1, 2}.
• The number of elements in a finite set A is its cardinality, written n(A).

There are two standard ways to write a set.

Roster form is best for small sets. Set-builder form is best for large or infinite sets, where listing is impossible.

• Null (empty) set: contains no element, written φ or { }. Example: {x : x is a natural number less than 1}.
• Singleton set: has exactly one element, such as {0}. Note that {0} is not empty — it contains 0.
• Finite and infinite sets: a finite set can be counted to an end ({1, 2, 3}); an infinite set cannot (the set of all natural numbers).
• Equal sets: A = B when they have exactly the same elements, e.g. {1, 2, 3} = {3, 2, 1}.
• Equivalent sets: have the same number of elements, n(A) = n(B), even if the elements differ: {a, b, c} and {1, 2, 3}.
• Disjoint sets: have no element in common, so A ∩ B = φ.
• Universal set (U): the set containing all elements under discussion in a problem.

A is a subset of B (A ⊆ B) if every element of A is also in B. A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.

• Every set is a subset of itself, and φ is a subset of every set.
• If a set has n elements, it has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
• The power set P(A) is the set of all subsets of A, so n(P(A)) = 2ⁿ.

Take U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

Useful laws to remember:

• Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A
• De Morgan's laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
• Counting formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)