Lecture - Introduction to University Mathematics, III

Notes

Some operations on sets:

• Union
• Intersection
• Complement, makes sense when $A$ is a subset of a larger set $S$. $A^C$ is every element in $S$ but not in $A$.
• The set difference $A \backslash B = {x\in A \,:\,x\notin B}$, every element in $A$ but not in $B$.

Two sets are disjoint if $A \cap B = \varnothing$ .

Double inclusion is the fact that $A = B \iff A \subseteq B \text{ and } B \subseteq A$. Can be proved by showing the left hand side implies the right hand side, and vice-versa. A pretty common proof technique for showing that two set expressions are equal.

Some distributive laws:

• $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
• $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

Can be proved by double inclusion.

De Morgan’s laws:

• $(A \cup B)^C = A^C \cap B^C$
• $(A \cap B)^C = A^C \cup B^C$

Can also be proved by double inclusion. De Morgan’s laws extend to more than just two sets.

An alternative method for proving set identities is via truth tables. The condition is “some arbitrary element is a member of this set”. Can also be used to prove De Morgan’s Laws.

One definition of cardinality, given recursively:

$\varnothing$ is finite and $ \vert \varnothing \vert = 0$. Any other finite set, $S$, has cardinality $ \vert S \vert = n+1$ if there exists an element $s \in S$ such that $ \vert S\backslash{s} \vert = n$ for some natural number $n$. Otherwise the set is infinite.

Induction works well for proving things about recursive definitions.

Flashcards