Sets Class 11: notes and important questions

Sets: revision notes

1. Sets and their representations

A set is a well-defined collection of objects (its elements); \(a \in A\) means \(a\) is an element of \(A\). Roster form lists the elements once each, in any order: \(\{2, 3, 5, 7\}\). Set-builder form states the property: \(\{x : x \text{ is a prime}, x \lt 10\}\).

2. Kinds of sets

• Empty set \(\phi = \{\,\}\): no elements (for example \(\{x \in \mathbb R : x^2 = -1\}\)).
• Finite set: a definite number of elements \(n(A)\); otherwise infinite (\(\mathbb N, \mathbb Z, \mathbb Q, \mathbb R\)).
• Equal sets: exactly the same elements. \(\{1, 2, 2, 3\} = \{3, 1, 2\}\).

3. Subsets and intervals

\(A \subset B\) if every element of \(A\) is in \(B\); \(A = B\) exactly when \(A \subset B\) and \(B \subset A\). For every set \(A\), both \(\phi\) and \(A\) itself are subsets of \(A\). Intervals are subsets of \(\mathbb R\):

• \((a, b) = \{x : a \lt x \lt b\}\), \([a, b] = \{x : a \le x \le b\}\), \([a, b) = \{x : a \le x \lt b\}\), \((a, b] = \{x : a \lt x \le b\}\); a round bracket leaves the end-point out.

4. Operations and Venn diagrams

• Union \(A \cup B = \{x : x \in A \text{ or } x \in B\}\); intersection \(A \cap B = \{x : x \in A \text{ and } x \in B\}\); \(A\) and \(B\) are disjoint if \(A \cap B = \phi\).
• Difference \(A - B = \{x : x \in A, x \notin B\}\).
• Laws: commutative, associative, \(A \cup \phi = A\), \(A \cap U = A\), distributive: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) and \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).
• A Venn diagram shows \(U\) as a rectangle and sets as circles; shade the region asked for.

5. Complement

\(A' = U - A = \{x \in U : x \notin A\}\). Properties: \(A \cup A' = U\), \(A \cap A' = \phi\), \((A')' = A\), \(\phi' = U\), \(U' = \phi\), and De Morgan's laws \((A \cup B)' = A' \cap B'\), \((A \cap B)' = A' \cup B'\). Also \(A - B = A \cap B'\).

Worked example 1

Write \(\{x : x \in \mathbb Z, x^2 \lt 10\}\) in roster form.

\(x^2 \lt 10\) for \(x = -3, -2, \ldots, 3\): \(\{-3, -2, -1, 0, 1, 2, 3\}\).

Worked example 2

Find \(A \cap B\) and \(A - B\) for \(A = [0, 4)\), \(B = (2, 6]\).

\(A \cap B = (2, 4)\), \(A - B = [0, 2]\).

Worked example 3

With \(U = \{1, \ldots, 8\}\), \(A = \{1, 2, 3\}\), \(B = \{3, 4\}\), find \((A \cup B)'\).

\(A \cup B = \{1, 2, 3, 4\}\), so \((A \cup B)' = \{5, 6, 7, 8\}\).

Common errors

• Repeating elements in roster form, or treating \(\{\phi\}\) as empty (it has one element).
• Getting the end-points of intervals wrong in intersections and differences.
• Writing \((A \cup B)' = A' \cup B'\): De Morgan's laws swap union and intersection.
• Confusing \(\in\) (element) with \(\subset\) (subset).

Exam tips

• For interval questions, draw both intervals on one number line first.
• To prove a set identity, show "\(x\) in the left side \(\Rightarrow x\) in the right side" and the reverse.