MATHSIGHT

A homomorphism is a function between two sets and such that the Group structure of the set is preserved. Let the set have a law of composition such that forms a Group . Let also the set have a law of composition Definition: Let     Then is a homomorphic map. It is important to note that the first is the law of composition on the set and the second is the law of composition on the set . If and and . If then under , . Then under…

Throughout this blog there are a number of symbols used that relate to mathematical logic. These are connectives such as and quantifiers . The connectives can be described by truth tables in logic. In summary: “From the proposition that something is true, it follows that it is true.”— Principia Mathematica is ‘AND’. is true if and only if both and are true. is ‘OR’. is true if either or or both and are true. is ‘IMPLIES’. is only false if is true and is false. is ‘IF AND ONLY IF’. is…

“Mathematics is the study of functions.”— Vito Volterra Theory Functions are evidently important in mathematics. The definition of a function follows from set theory and it is worth exploring the derivation. Commonly a function is seen as something like . However, their definition is grounded in set theory. A function should be seen as an assignment of an element of a set to an element of a set with some restrictions. This idea can be stated as:     Where is the function. The image of is the element in the…

I am a structural engineer that has spent a lifetime of using numbers to make buildings stand up and sometimes, to explain why they don’t! I enjoying seeing beauty in the world through striving to understand the analytical principles that underpin our life. As a hobby, I like to read mathematics and physics to underpin a deeper understanding of the analysis in the world. Rather than keep this learning to myself, I chose to inflict it on the world in this blog as motivation to keep learning. I first enjoyed…