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Mathematics Formula Handbook

Essential Formula Sheets

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Chapter 01•9 High-Yield Equations

Sets

Practice PYQs
#01

Subset

A⊆B⇔∀x∈A,  x∈BA \subseteq B \Leftrightarrow \forall x \in A,\; x \in B
Every element of A is in B
#02

Power Set Cardinality

∣P(A)∣=2n where n=∣A∣|P(A)| = 2^n \text{ where } n = |A|
Number of subsets of A
#03

Union

A∪B={x:x∈A or x∈B}A \cup B = \{x : x \in A \text{ or } x \in B\}
All elements in A or B or both
#04

Intersection

A∩B={x:x∈A and x∈B}A \cap B = \{x : x \in A \text{ and } x \in B\}
Elements common to both A and B
#05

Complement

A′=U−A={x:x∈U,  x∉A}A' = U - A = \{x : x \in U,\; x \notin A\}
Elements in universal set but not in A
#06

De Morgan's Laws

(A∪B)′=A′∩B′,(A∩B)′=A′∪B′(A \cup B)' = A' \cap B',\quad (A \cap B)' = A' \cup B'
Complement distributes over union/intersection
#07

Inclusion-Exclusion (2 sets)

∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|
Avoids double counting
#08

Inclusion-Exclusion (3 sets)

∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣B∩C∣−∣A∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|
Generalization for 3 sets
#09

Cartesian Product

A×B={(a,b):a∈A,  b∈B},∣A×B∣=∣A∣⋅∣B∣A \times B = \{(a,b) : a \in A,\; b \in B\},\quad |A\times B| = |A|\cdot|B|
Set of all ordered pairs
1 of 29 Chapters

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