Sets
Symmetric Difference (△) - The symmetric difference of two sets is the collection of elements which are members of either set but not both - in other words, the union of the sets excluding their intersection. Forming the symmetric difference of two sets is simple, but forming the symmetric difference of three sets is a bit trickier
Ex - Given two sets (for example set A = 3 and set B = 4), the mathematical term "symmetric difference" of two sets is the set of elements which are in either of the two sets, but not in both (A △ B = C = 4).
Bijection, injection and surjection
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments(input expressions from the domain) and images(output expressions from the codomain) are related or mapped to each other.
A function maps elements from its domain to elements in its codomain. Given a functionf: X -> Y- The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection. Notationally:
Or, equivalently (using logical transposition),
- The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal. A surjective function is a surjection. Notationally:
- The function is bijective(one-to-one and onto or one-to-one correspondence) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective. A bijective function is also called abijection.
An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.
https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
Cartesian product
In mathematics, the Cartesian product of two sets is the set of all possible ordered pairs. The first element of each pair comes from the first set, and the second element comes from the second set. The Cartesian product is denoted by A×B.
For example, if A = 2 and B = 5, then the Cartesian product of A and B is 5.
The Cartesian Product of sets A and B is defined as the set of all ordered pairs (x, y) such that x belongs to A and y belongs to B.