Notes On Set Theory.docx

Notes on SET THEORY:

1. A set S is made up of elements and if α is one of these element, we shall denote this fact by α ε S. 2. There is exactly one set with no elements. It is the empty set and is denoted by ∅. 3. There are two ways by which a set maybe described: roster or setbuilder notation. Roster method is the listing method, while the set-builder method uses symbols. 4. A set S is well-defined means that if S is a set and α is some object then either α is definitely in S or α is definitely not in S. 5. A set B is a subset of set A, denoted by 𝐵 ⊆ 𝐴 𝑜𝑟 𝐴 ⊇ 𝐵. 6. If B is a subset of A where 𝐵 ≠ 𝐴, then we write 𝐵 ⊂ 𝐴. 7. If A is any set, then A is the improper subset of A. Any other subset of A is a proper subset of A. 8. The cardinality of a set X is the number of elements in the set denoted by |𝑋|. 9. Let A and B be sets, the set 𝐴 𝑥 𝐵 = {(𝑎, 𝑏)⁄𝑎 𝜀 𝐴, 𝑏 𝜀 𝐵 }. 10.A relation between sets A and B is a subset ℜ of A x B. We read (𝑎, 𝑏) ℇ ℜ as “ 𝑎 is related to b” or 𝑎 ℜ 𝑏. 11. A function ∅ mapping X into Y is a relation between X and Y with the property that each 𝑥 𝜖 𝑋 appears as the first member of exactly one ordered pair (x,y) in ∅. Such a function is also called a map or mapping X into Y . We write ∅ ∶ 𝑋 → 𝑌 𝑎𝑛𝑑 (𝑥, 𝑦)𝜀 ∅ 𝑚𝑒𝑎𝑛𝑠 𝑦 = ∅ (𝑥). 12. The domain of ∅ is the set X.

13. The codomain of ∅ is the set Y. 14. The range of ∅ is the set ∅ (𝑥) = {∅(𝑥)/ 𝑥 𝜀 𝑋} 15. A function ∅ ∶ 𝑋 → 𝑌 is one-to-one (injection) if ∅ (𝑥) = ∅ (𝑦) only when x = y. 16. A function ∅ ∶ 𝑋 → 𝑌 is onto (surjection) if the range of ∅ 𝑖𝑠 𝑦. 17. A function is bijective (bijection) if it is both one-to-one and onto. 18. Two sets X and Y have the same cardinality if there exists a 1-1 function mapping X onto Y. 19. A partition of a set S is a grouping of S into non-empty disjoint subsets such that every element of S is in exactly one of the subsets. The subsets are the cells of the partition. [𝑥] is the cell where the element is found. 20.Each partition of a set S yields a relation ℛ on S in natural way namely: For x , y 𝜀 S, x ℛ 𝑦 iff x and y are the same cell of the partition, This relation ℛ of S satisfies the properties of equivalence relations. 21. An equivalence relation ℛ on a non-empty set S satisfies in the following: (1) reflexive property - 𝑥ℛ𝑥, ∀ 𝑥 ℰ 𝑆; (2) symmetric property – If 𝑥ℛ𝑦 , 𝑡ℎ𝑒𝑛 𝑦ℛ𝑥 for x, y ℰ 𝑆 ; (3) transitive property – If 𝑥ℛ𝑦 , 𝑦ℛ𝑧, 𝑡ℎ𝑒𝑛 𝑥ℛ𝑧 for x, y, z ℰ 𝑆

A “function” is a rule which maps each element of a first set, called the “domain” of the function, to one element of a second set, called the “codomain” of the function.

For a function f:A->B, A is the “domain” (set which is “mapped-from”) and B is the “codomain” (set which is “mapped-to”). If S is a subset of A, then the “image” of S under f is that subset T of B consisting of all the points in B which f actually does map-to from points in A. The image of the entire domain is called the “range” of f. If the range is the same as the codomain, then the function is “surjective” or “onto”. An example would be g:R->R, y=g(x)=x3x3. All points of R are mapped-to by g, so the range is the codomain and g is “surjective”. However, the function h:R->R, y=h(x)=x4x4 is not surjective, because h maps R onto R+0R0+ only; none of the values of x4x4 are negative. There is also the concept of “injective” or “one-to-one”; if no two elements of A are mapped to the same element of B by a function f:A->B, then f is “injective”. Continuous functions from R to R which are strictly-increasing or strictly-decreasing are always injective. Functions which are both “surjective” and “injective” are called “bijective”. This is a very useful concept for multiple reasons, not the least of which are these two: Firstly, a function is invertible if-and-only-if it is bijective. And secondly, two sets A and B have the same cardinality (size) if-and-only-if a bijection f exists between them. 1

Be a function f:X→Yf:X→Y Then for every x there is a value f(x) in the codomain, but not neccesary all the f(x) are in the codomain (or range). The set of all f(x) are the image x²:ℝ→ℝ has the co-domain ℝ but the image (for the whole domain) [0, +∞)