This chapter introduces the language of sets — the notation, definitions, and operations you need to work with collections of objects. This sheet covers the key formulas: set-builder and roster forms, subset and equality conditions, union, intersection, difference, complement, interval notation, and De Morgan’s laws.
Each formula is grouped by topic with the meaning of every symbol, guidance on when to use it, and original worked examples. For the detailed explanations and derivations, visit the
Answer: De Morgan’s law is verified. \(A’ = \{1,3,5,7,9\}\), \(B’ = \{2,4,6,8,9\}\).
Example 3: Interval Notation and Set-Builder Form
Step 1: Write the set \(\{x : x \in \mathbf{R}, -3 \leq x \lt 5\}\) as an interval.
Step 2: The left endpoint \(-3\) is included (≤), the right endpoint \(5\) is excluded ().
So the interval is \([-3,5)\).
Step 1: Write the interval \((2,7]\) in set-builder form.
Step 2: \(2\) is excluded (open), \(7\) is included (closed).
So \(\{x : x \in \mathbf{R}, 2 \lt x \leq 7\}\).
Answer: \([-3,5)\) and \(\{x : 2 \lt x \leq 7\}\).
Common Mistakes to Avoid
Mistake
Correct rule
How to check your answer
Writing \(a \in \{a,b\}\) but then saying \(\{a\} \in \{a,b\}\) (confusing element and subset).
\(a \in A\) means a is an element; \(\{a\} \subset A\) means the set containing a is a subset.
Ask: is the object on the left an element of the set? If it is a set itself, use \(\subset\) not \(\in\).
Forgetting that the empty set is a subset of every set.
\(\phi \subset A\) for any set A.
When listing subsets, always include \(\phi\).
Mixing up union and intersection when taking complements, e.g., writing \((A \cup B)’ = A’ \cup B’\).
De Morgan’s law: \((A \cup B)’ = A’ \cap B’\).
Test with a small example (like Example 2 above) to verify.
Writing an interval \((a,b)\) when the endpoints are included, or \([a,b]\) when they are excluded.
Use round brackets \(( \) for strict inequalities (\(\lt , \gt\)), square brackets \([ \) for inclusive (\(\leq , \geq\)).
Check the inequality signs in the set-builder form.
Assuming a set with repeated elements is different from the set without repetitions.
A set is defined by its distinct elements; repetitions do not change the set.
Rewrite the set without duplicates and compare.
Frequently Asked Questions
What is the difference between \(\subset\) and \(\in\)?
\(\subset\) is a relation between two sets: \(A \subset B\) means every element of A is also in B. \(\in\) is a relation between an element and a set: \(a \in A\) means a is an element of A. For example, \(2 \in \{2,3\}\) but \(\{2\} \subset \{2,3\}\).
How do I know if a set is finite or infinite?
A set is finite if it is empty or has a definite number of elements (you can count them). It is infinite if its elements cannot be listed completely (e.g., the set of natural numbers, the set of points on a line). Look at the definition: if the set has no bound (like “all even numbers”) it is infinite.
What is the complement of a set?
The complement \(A’\) of a set A is the set of all elements in the universal set U that are not in A. It is the same as \(U – A\). The complement depends on the choice of U.
When do I use De Morgan’s laws?
Use them when you need to find the complement of a union or intersection. They let you “distribute” the complement: the complement of a union is the intersection of the complements, and vice versa. This is useful for simplifying expressions in set theory.
Practise these formulas on the textbook’s own questions — see the
