Week 1- Fundamentals of Sets and Functions

These flashcards cover key concepts related to sets, functions, and their properties as discussed in the lecture notes.

Set

A collection of distinct objects, considered as an object in its own right.

Finite Set

A set with a limited number of elements.

Infinite Set

A set that has no end or limit to the number of elements.

Cartesian Product

A set of all ordered pairs (a, b) formed by combining each element of one set with each element of another set.

Set Intersection

The set of all elements that are common to both sets, denoted as S ∩ T.

Set Union

The set of all elements that are in either set or both sets, denoted as S ∪ T.

Set Difference

The set of elements that are in one set but not in another, denoted as S \ T.

Subset

A set A is a subset of set B if every element of A is also an element of B, denoted as A ⊂ B.

Injective Function (One-to-One)

A function where different inputs always produce different outputs.

Surjective Function (Onto)

A function where every element in the target set is mapped to by at least one element from the domain.

Bijective Function

A function that is both injective and surjective; each element of the target set is paired with exactly one element of the domain.

Quantifier ∀ (For all)

A symbol used in logic that indicates that a statement applies to all elements of a specified set.

Quantifier ∃ (There exists)

A symbol used in logic to indicate that there is at least one element in a specified set for which a statement is true.

Function

A special type of relation that assigns each element of a set exactly one element of another set.

Function Composition

The process of combining two functions, where the output of one function becomes the input of another.

Empty Set

A set that contains no elements, denoted by the symbol ∅.

Real Numbers (R)

The set of all rational and irrational numbers.

Complex Numbers (C)

The set of numbers that includes all real numbers and imaginary numbers, represented as a + bi where a and b are real numbers.

Natural Numbers (N)

The set of positive integers commonly used for counting.

Integers (Z)

The set of whole numbers that can be positive, negative, or zero.

Continuous Functions

Functions that do not have any breaks, jumps, or holes in their graphs.

Polynomial Functions

Functions that can be expressed in the form of a polynomial, e.g., P(x) = anx^n + a(n-1)x^(n-1) + … + a_0.