OT1: Mathematical argument, language and proof: Understand and use language and symbols associated with set theory, as set out in the appendices. Apply to solutions of inequalities and probability.

The idea

Set language gives a precise way to describe collections of values and events. You need membership (∈, ∉), subset (⊆) and proper subset (⊂), union (∪), intersection (∩), complement (′ or bar), the empty set (∅), the universal set (ξ), number sets (ℕ, ℤ, ℚ, ℝ) and the number of elements n(A). In inequalities, the solution set is a subset of ℝ: 2 < x ≤ 5 is the interval (2, 5], and solving two inequalities joined by 'and' means intersecting their solution sets, while 'or' means taking the union. In probability, events are subsets of the sample space; P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and the complement rule P(A′) = 1 − P(A) follows from set logic. Always define the universal set before using complements, and translate accurately between words, symbols and number-line or Venn diagrams.

What earns the marks

• Correctly uses ∈ and ∉ to state whether a given number or outcome belongs to a defined set.
• Uses ⊆, ⊂, ∪, ∩ and ′ (or bar notation) accurately when combining sets or events, including recognising when one set is a subset or proper subset of another.
• Uses the universal set ξ, the empty set ∅, number sets ℕ, ℤ, ℚ, ℝ and the notation n(A) for the number of elements in a finite set.
• Translates an inequality solution into interval notation, choosing open or closed brackets correctly for strict and non-strict inequalities.
• Solves a pair of simultaneous inequalities by intersecting solution sets and expresses the result using set or interval notation.
• Applies the addition law P(A ∪ B) = P(A) + P(B) − P(A ∩ B) and the complement rule P(A′) = 1 − P(A) to probability problems.
• Interprets Venn diagrams and number-line diagrams, including identifying regions representing unions, intersections and complements.

Common mistakes and how to put them right

• Writing (2, 5) when the inequality is 2 < x ≤ 5.

  Put it right: The correct interval is (2, 5] because the upper endpoint is included.

• Treating 'and' and 'or' as interchangeable when combining inequalities.

  Put it right: 'and' gives an intersection, 'or' gives a union, and the resulting sets differ.

• Using the complement without stating the universal set, so A′ is ambiguous.

  Put it right: Identify the sample space or universal set first.

• Adding P(A) and P(B) without subtracting P(A ∩ B) when events overlap, which double-counts the overlap.

• Confusing ⊆ with ⊂.

  Put it right: A ⊆ A is true, but A ⊂ A is false because a proper subset must be strictly smaller.

Exam tips

• Write the universal set or sample space explicitly before using complement notation, so your working is unambiguous.
• Sketch a number line for inequalities and a Venn diagram for probability events; these make intersections, unions and complements visible and reduce errors.
• Check interval brackets against the inequality symbols: strict inequalities take round brackets, non-strict take square brackets.
• When a question uses set notation, answer in the same notation unless told otherwise, and show the intermediate set before giving the final interval or probability.