DPST1012 Chap.1 Sets, Intervals and Algebra Skills
Sets, Intervals and Algebra Skills
Why the course starts with sets
The first week of Fundamentals of Mathematics B is labelled Assumed Knowledge, and the syllabus gives it four jobs: set notation, union and intersection, the addition and complement rules, and interval notation. These are not side topics.
A domain is a set, a solution to an inequality is an interval, and an event in the probability weeks is a set of outcomes, so the symbols learned here are reused in almost every later chapter. The Week 4 test assesses this material alongside functions and polynomials.
Counting without double counting
The central skill is the addition rule.
When two groups overlap, their sizes cannot simply be added, because the shared members would be counted twice; subtracting the intersection fixes that. The complement rule then answers the opposite question, how many are left outside. Survey problems with three overlapping groups extend the same idea: subtract every pairwise overlap, then add back the members who belong to all three.
A three-circle diagram filled from the centre outwards is the safest way to set these out.
Intervals and the algebra toolkit
Interval notation is a compact way to write a set of real numbers. Square brackets keep an endpoint and round brackets leave it out, and infinity always takes a round bracket because it is not a number that can be included.
Alongside the set work, a Basic Skills worksheet revises linear equations, rearranging formulas, inequalities, factorising, algebraic fractions, simultaneous equations, index laws and surds. The one rule that catches most students is that multiplying or dividing an inequality by a negative number reverses its direction.
What this chapter covers
- 01
Sets, elements and set-builder notation
- 02
Subsets, the universal set and complements
- 03
Union, intersection and disjoint sets
- 04
Addition rule and complement rule
- 05
Three-set survey counting
- 06
Interval notation and rays
- 07
Index laws, factorising and surds
Worked example · free
Count the members of no group in a three-way survey
- 1Add the three group sizes: 40 + 35 + 30 = 105.
- 1Subtract the pairwise overlaps: 105 − (12 + 10 + 8) = 75.
- 1Add back the members in all three, who were removed once too often: 75 + 3 = 78.
- 1Use the complement rule: 90 − 78 = 12 members do none of the three.
Key terms
- Universal Set
- The set U of every object under discussion in a problem, from which complements are taken.
- Complement
- The set of elements of U that are not in A, written A with a superscript c.
- Disjoint Sets
- Two sets with no element in common, so their intersection is the empty set.
- Closed Interval
- A set of real numbers between two endpoints that includes both endpoints, written with square brackets.
- Power Set
- The collection of all subsets of a set; a set with n elements has 2 to the power n of them.
Sets, Intervals and Algebra Skills FAQ
Why does the addition rule subtract the intersection?
Because adding the two group sizes counts every shared member twice, once in each group. Taking away the size of the overlap once leaves each member counted exactly one time, which is what the size of the union means.
Is an interval with a square bracket at infinity ever correct?
No. Infinity is not a real number, so it can never be included as an endpoint. Rays are always written with a round bracket at the infinite end, such as [2, ∞) for all real numbers from 2 upwards.
What makes two sets equivalent rather than equal?
Equivalent sets only need the same number of elements, whatever they are. Equal sets need exactly the same elements. So a set of three fruits and a set of three digits are equivalent but not equal.
How much of the basic algebra revision is examined later?
All of it reappears inside later topics rather than as separate questions. Factorising drives polynomial roots, index laws drive exponentials and derivatives, and inequality rules return whenever a solution is a range.
How do I check an interval answer quickly?
Pick one number inside your interval and one just outside it, and substitute both into the original inequality or set description. The inside number should satisfy it and the outside number should not; if either fails, a bracket or a direction is wrong.
Exam move
Rebuild each set rule from a small example of your own before using it on a survey question, and test every inequality answer by substituting one value from inside and one from outside the interval. Ten minutes of this before the Week 4 test is worth more than an hour of rereading.
Working through Sets, Intervals and Algebra Skills in DPST1012? Sia is AskSia’s AI Mathematics tutor — ask any DPST1012 Sets, Intervals and Algebra Skills question and get a clear, step-by-step explanation grounded in how DPST1012 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
