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Formula Sheets / Maths

Chapter 13

Sets and Relations

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Sets

Union and intersection
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
Three sets
$n (A \cup B \cup C) = \sum n (A) - \sum n (A \cap B) + n (A \cap B \cap C)$
De Morgan
$(A \cup B)^{'} = A^{'} \cap B^{'},$ $(A \cap B)^{'} = A^{'} \cup B^{'}$
Difference
$A - B = A \cap B^{'},$ $n (A - B) = n (A) - n (A \cap B)$
Power set
$n (A) = m \Rightarrow n (P (A)) = 2^{m}$

Relations

Number of relations
$n (A) = m, n (B) = n : 2^{mn}$ relations from $A$ to $B$
Types
reflexive: $(a, a) \in R \forall a;$ symmetric: $(a, b) \in R \Rightarrow (b, a) \in R;$ transitive: $(a, b), (b, c) \in R \Rightarrow (a, c) \in R$
Counts on a set of n elements
reflexive: $2^{n^{2} - n},$ symmetric: $2^{\frac{n ( n + 1 )}{2}}$
Equivalence relation
reflexive $+$ symmetric $+$ transitive

Solutions of Triangles Functions