Sets, Functions and Groups
Chapter Summary
This chapter forms the foundational bedrock of discrete mathematics and abstract algebra, heavily tested in both short and long questions in BISE board exams. You must master the basic algebra of sets, including operations like union, intersection, difference, and the complement of a set. A major exam focus is the rigorous application of Venn diagrams and De Morgan’s Laws ($$(A \cup B)' = A' \cap B'$$ and $$(A \cap B)' = A' \cup B'$$), which frequently appear in both proof-based questions and practical word problems. Ensure you thoroughly practice proving set identities using logical equivalences and subset inclusions.
Moving into relations and functions, board examiners frequently test your understanding of domain, co-domain, and range. You need to clearly distinguish between injective (one-to-one), surjective (onto), and bijective functions, as proving whether a given function is bijective is a staple long-question item. Additionally, expect numerical problems involving inverse functions ($f^{-1}$) and the composition of functions ($fog$ and $gof$), where algebraic manipulation and finding restricted domains to make a function invertible are crucial skills.
The final and most scoring section introduces groups and abstract algebraic structures. To prove a given set with a binary operation is a group, you must systematically verify four axioms: closure, associativity, the existence of an identity element, and the existence of an inverse element for every member. If the operation is commutative, it forms an Abelian group. Pay special attention to finite groups under modulo arithmetic (e.g., multiplication or addition modulo $n$) and the properties of cube roots of unity, as examiners routinely set these for 4-mark or 8-mark analytical questions.
Important Questions
- Use truth tables to prove the distributive law: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.5 marks
- Prove that $(A \cup B)' = A' \cap B'$ (De Morgan's Law) using set-builder notation.5 marks
- If $f: X \to Y$ is a function, prove that $f$ is injective if and only if for all subsets $A, B \subseteq X$, $f(A \cap B) = f(A) \cap f(B)$.5 marks
- Let $f: A \to B$ and $g: B \to C$ be two functions. If the composite function $g \circ f: A \to C$ is injective, prove that $f$ is injective.5 marks
- Show that the set of all matrices of the form $\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$ where $a \in \mathbb{R} \setminus \{0\}$ forms an abelian group under matrix multiplication.10 marks
- Prove that the 4th roots of unity ($1, -1, i, -i$) form an abelian group under ordinary multiplication.5 marks
- Find the inverse of the function $f(x) = \frac{x+1}{x-1}$ for $x \neq 1$, and show that $f^{-1} = f$.3 marks
- Define a binary operation on $\mathbb{Q}$ by $a * b = a + b - ab$. Determine whether $\mathbb{Q}$ is a monoid under this operation.5 marks