Mathematics: Sets and Operations

Mathematics

• Sets: A set is a well-defined collection of distinct objects. Elements are the objects within a set. Sets are often denoted by capital letters, elements by lowercase letters. Sets can be described by listing their elements (roster method) or by a rule that defines the elements (set-builder notation).
• Set Operations:
  • Union: The union of two sets (A ∪ B) contains all elements in set A or set B or both.
  • Intersection: The intersection of two sets (A ∩ B) contains only the elements common to both set A and set B.
  • Difference: The difference of set A and set B (A \ B) contains the elements in A but not in B.
  • Complement: The complement of a set A (A^c) contains all elements in the universal set (containing all relevant elements) that are not in A.
• Types of Sets:
  • Empty Set: A set with no elements (denoted by ∅ or {}).
  • Finite Set: A set containing a fixed number of elements.
  • Infinite Set: A set containing an unlimited number of elements.
  • Universal Set: A set containing all elements relevant to a given discussion.
  • Subset: Set A is a subset of set B (A ⊆ B) if every element of A is also an element of B.
  • Proper Subset: Set A is a proper subset of set B (A ⊂ B) if A is a subset of B but they are not equal (A ≠ B).

Limits

• Limits of Functions: The limit of a function as x approaches a value 'a' is the value that the function approaches as x gets arbitrarily close to 'a'. If the limit exists, this means the function approaches a single value.
• Formal Definition of Limit:
  • If for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This means for any small error ε in the output, we can find a small enough interval around a, such that all the inputs in that interval have a corresponding output within the tolerance ε of the value L.
• One-sided Limits: Limits can be considered as x approaches 'a' from the left (lim_x→a⁻ f(x)) or from the right (lim_x→a⁺ f(x)).
• Calculating Limits (Direct Substitution, Factorization, Rationalization): Methods used to evaluate limits algebraically.

Derivatives

• Derivative of a Function: The derivative of a function at a point represents the instantaneous rate of change of the function at that point. It is the slope of the tangent line to the curve at that point.
• Definition of the Derivative: The derivative of a function f(x) at a point x = a is given by f'(a)=lim_h→0[f(a+h) - f(a)]/h. It measures the instantaneous rate of change (slope) in f(x) as 'x' changes very slightly.
• Rules of Differentiation: Methods for finding derivatives of various functions:
  • Power Rule (d/dx (xⁿ) = nx^n-1).
  • Constant Multiple Rule (d/dx (cf(x)) = c(d/dx(f(x)))).
  • Sum/Difference Rule (d/dx (f(x) ± g(x)) = (d/dx (f(x)) ± (d/dx (g(x)))).
  • Product Rule (d/dx (f(x)g(x)) = f(x)g'(x) + g(x)f'(x)).
  • Quotient Rule (d/dx (f(x)/g(x)) = [g(x)f'(x) - f(x)g'(x)]/[g(x)]²).
  • Chain Rule (d/dx(f(g(x))) = f'(g(x))g'(x)).
• Higher-Order Derivatives: Successive derivatives, like the second derivative f"(x) or third derivative f'''(x).