Formula Sheet: Sets
This chapter introduces the concept of sets, their significance, and basic operations in mathematics.
Sets – Formula & Equation Sheet
Essential formulas and equations from the Sets chapter of Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Sets chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
A ∪ B = {x : x ∈ A or x ∈ B}
The union of sets A and B is the set of elements that are in either A or B (or both). Useful for combining different collections.
A ∩ B = {x : x ∈ A and x ∈ B}
The intersection of sets A and B consists of elements that are in both A and B. This is crucial for identifying common elements.
A - B = {x : x ∈ A and x ∉ B}
The difference of sets A and B includes elements that are in A but not in B. This is used to find what is unique to a set.
A' = {x : x ∈ U and x ∉ A}
The complement of set A includes all elements in the universal set U that are not in A. Helps in understanding what is excluded from A.
φ = {}, A ⊆ B if every x ∈ A implies x ∈ B
The empty set (φ) has no elements. A is a subset of B if all elements of A are also elements of B. Important for defining relationships between sets.
A = B if A ⊆ B and B ⊆ A
Two sets A and B are equal if they contain the same elements. Essential for proving set equivalences.
n(S) = number of distinct elements in set S
The notation n(S) denotes the number of elements in a set S. This is important for calculating cardinality.
n(φ) = 0
The cardinality of the empty set is zero because it contains no elements. Fundamental in set theory.
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
This relation calculates the number of elements in the union of two sets by adding their sizes and subtracting the size of their intersection to avoid double counting.
De Morgan's Laws: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'
These laws describe the relationship between union/intersection and complements, helping in manipulating set expressions.
Equations
A = {x : x is an even natural number}
This set-builder notation defines A as containing all even natural numbers. Used for systematic definition of sets.
B = {x : x is a prime number}
Set B consists of all prime numbers. Critical for studies involving number theory and properties of numbers.
C = {x : x < 5, x ∈ Z}
This defines set C as all integers less than 5. Set-builder notation helps clarify conditions for membership.
D = {x : x is a digit from 0 to 9}
Set D is defined to include all single-digit natural numbers. Essential for understanding numerical sets.
{x : x ∈ N, x > 3}
This defines a set of natural numbers greater than 3, demonstrating how to set conditions for membership.
E = {x : x^2 ≤ 25}
Set E includes all numbers whose square is less than or equal to 25. Useful for quadratic inequalities.
F = {x : x = 3n, n ∈ N}
Set F represents multiples of 3. Important in understanding arithmetic sequences.
G = {x : x ∈ R, x is negative}
This defines set G to include all negative real numbers, useful for interval studies.
H = {x : x is a letter in the word 'MATH'}
Set H contains specific letters. Demonstrates practical applications of set definitions.
I = {x : x = n^2, n ∈ Z}
Defines set I as containing perfect squares of integers. Important in algebra and number theory.
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