Mathematics I · Week 1 — Sets, relations, and functions

Week map

Set notation (learn the symbols before the rules)

• $A$ → “set $A$” → collection of distinct objects → $A = \{2, 5, 7\}$.
• $\{x : \text{condition}\}$ → “set builder” → all $x$ satisfying the condition → $\{x : x \in \mathbb{Z}, -2 < x < 4\} = \{-1,0,1,2,3\}$.
• $x \in A$ → “$x$ belongs to $A$” → membership test → $3 \in \{1,3,5\}$ is true.
• $A \subseteq B$ → “$A$ is a subset of $B$” → every element of $A$ is in $B$ → $\{2,4\} \subseteq \{1,2,3,4\}$.
• $A \cup B$ → “union” → elements in $A$ or $B$ (or both) → $\{1,2\} \cup \{2,3\} = \{1,2,3\}$.
• $A \cap B$ → “intersection” → elements in both → $\{1,2\} \cap \{2,3\} = \{2\}$.
• $A^c$ or $A'$ → “complement” → elements in universal set $U$ not in $A$ → if $U=\{1,2,3,4\}$, then $\{1,3\}^c = \{2,4\}$.
• $|A|$ → “cardinality” → number of elements → $|\{a,b,c\}| = 3$.

Roster vs interval on the line

• $(a,b)$ → open interval → $a < x < b$ → $(2,5)$ excludes endpoints.
• $[a,b]$ → closed interval → $a \le x \le b$.
• $(-\infty, b]$ → unbounded left → all $x \le b$.

Relations

Relation properties (when $R$ is on $A \times A$)

• Reflexive: every element pairs with itself → $(a,a) \in R$ for all $a \in A$.
• Symmetric: swapping order keeps membership → if $(a,b) \in R$ then $(b,a) \in R$.
• Transitive: chains close → if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$.

Functions

• Range → actual outputs hit → subset of codomain.
• $f(x)$ → value at $x$ → unique output rule.

Injective, surjective, bijective

• Injective (one-to-one): different inputs → different outputs. Fail if two inputs share an output.
• Surjective (onto): every codomain element is hit. Fail if some $b \in B$ never appears.
• Bijective: both injective and surjective — inputs and outputs pair perfectly.

Pattern families

Easy — Membership and operations

• List elements after union/intersection/complement with universal set stated.
• Convert roster to set-builder when pattern is clear (even integers between 0 and 10).
• Count $|A|$ after combining sets; use inclusion when asked “how many in $A$ or $B$ but not both”: $|A \cup B| - |A \cap B|$ for symmetric difference size.

Medium — Subset and power set

• List all subsets of a small set; confirm $2^n$ count for $n$ elements.
• Decide whether $X \subseteq Y$ from explicit rosters or interval containment.
• Given $U$ and $A$, describe $A^c$ in roster form.

Hard — Relation vs function and mapping type

• From a table of pairs, mark function or not; identify domain and range.
• Decide injective/surjective from arrow diagram or table without computing formula.
• Combine set ops with function definition: e.g. domain restricted to $A \cap B$.

Worked mini-examples

Traps

• Confusing $\in$ (element) with $\subseteq$ (subset). $\{1\} \subseteq \{1,2\}$ but $\{1\} \notin \{1,2\}$ as an element (unless nested).
• Forgetting $\emptyset$ is subset of every set.
• Calling a relation a function when one input has two outputs in the table.
• Range vs codomain: codomain is declared; range is what actually occurs.
• Power set of $\{a\}$ is $\{\emptyset, \{a\}\}$ — two elements, not one.

Diagnostic (try yourself)

1. Let $U = \{1,2,3,4,5,6\}$, $P = \{2,4,6\}$, $Q = \{1,2,3\}$. List all elements of $(P \cup Q)^c$.
2. How many subsets does $\{w,x,y,z\}$ have? How many of those subsets contain $w$?
3. A relation on $\{0,1,2\}$ is given by pairs $\{(0,0),(1,0),(1,1),(2,2)\}$. Is it a function from $\{0,1,2\}$ to $\{0,1,2\}$? If not, which input fails?
4. Define $f : \{1,2,3,4\} \to \{5,6,7\}$ by $f(1)=5, f(2)=6, f(3)=7, f(4)=5$. Is $f$ injective? Surjective? Bijective?
5. Express in roster form: $\{x \in \mathbb{Z} : x^2 < 10\}$.