Outrageously Funny Word Dictionary :: Xylogics

👉 The term "xylogics" refers to a branch of mathematics known as algebraic logic, which deals with the properties of real-valued functions that are defined on sets or graphs. In algebraic logic, one considers the operation of applying functions defined on sets or graphs (such as addition, multiplication, and composition) to specific values of these functions. This is done through a set of axioms known as the "axiom system" for algebraic logic. Some key properties of real-valued functions include: 1. Identity: A function $f : X \rightarrow Y$ has an identity if it satisfies $f(x+y) = f(x) + f(y)$ and $f(x) = 0$ if $x=0$. This means that the sum or product of two functions is equal to the identity function. 2. Inverses: Given a function $f : X \rightarrow Y$, its inverse (or reciprocal, adjunction or reflection) can be found by swapping inputs and outputs: $f^{-1}(y) = x \quad \text{if} \quad f(x) = y$ 3. Distributivity of Functions: For two functions $f,g$ on the same set $X$ and any real numbers $a,b$, we have that: $$f(a+b) = f(a) + f(b)$$ 4. Associativity: Given three functions $f,g,h$, it is true that $(f \circ g)(x) = f(g(x))$ 5. Commutativity (Transitivity): For any function $f : X \rightarrow Y$ and any two elements of the set $X$, we have: $$f(x) = x \quad \text{if} \quad f(y) = y$$ 6. Absorption Laws: Given a real number $b$ and functions $f,g$, it is true that $(g \circ f)(x) = g(f(x))$ 7. Continuity: A function is continuous at a point if the value of the function at that point does not change with respect to its argument. In algebraic logic, one seeks properties of real-valued functions that hold in many possible ways, so as not to restrict any particular application or analysis. These axioms and properties are fundamental for understanding how real-valued functions behave in various contexts.

xylogics