How is this so easy to understand when u say it, retook my uni exam 2 times now watched your video & it finally makes so much sense you need to train lecturers honestly! So good
I'm sorry that you've struggled with your exams, but I'm happy that you understand things now. Honestly, I think that many lecturers don't spend enough time on examples that illustrate the concepts. Even though the course is *abstract* algebra, the ideas grew out of looking at specific cases. It is usually easier to see the properties "at work" than trying to figure things out just from the definitions.
Before (a priori, group) is dual to after (a posteriori, image) -- Immanuel Kant. Normal subgroups are dual to homomorphism (factor groups) synthesize the kernel. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Being is dual to non-being creates becoming -- Plato. Domains (groups) are dual to codomains (image, range). Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point, non null homotopic requires at least two points (duality) -- topology. Polar opposites of the dyad unite into one or the monad - opposame. "Always two there are" -- Yoda.
@@maxpercer7119 Increasing the number of states or dimensions is an entropic process -- co-homology. Decreasing the number of states or dimensions is a syntropic process -- homology. Homology (convergence, syntropy) is dual to to co-homology (divergence, entropy) -- topology. The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking. Your mind is syntropic as it makes predictions to track targets and goals -- teleology. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line. In a communication system the receiver of a message literally predicts the message into existence -- Shannon information theory. There is a dual process to that of increasing entropy. In information theory average information (entropy) becomes mutual information (syntropy) -- duality! Y = X. Y is an equivalent or dual description of X, all mathematical equations are dualities. Main stream physics is currently dominated by teleophobia or eliminative materialism. "Philosophy is dead" -- Stephen Hawking. Teleological physics (syntropy) is dual to non teleological physics (entropy). "Always two there are" -- Yoda. Addition is dual to additive inverses (subtraction) -- Abstract algebra. Multiplication is dual to multiplicative inverses (division) -- Abstract algebra. Integration (syntropy) is dual to differentiation (entropy). Elliptic curves are dual to modular forms. Rational, analytic (mathematics) is dual to empirical, synthetic (physics) -- Immanuel Kant. Noumenal (a priori) is dual to phenomenal (a posteriori) -- Immanuel Kant. There are patterns of duality hardwired into physics, mathematics and philosophy!
It would honestly be a very boring video. Assuming the group operations are addition (with respect to the appropriate mod) in each group, the only homomorphism is the "trivial" one that maps every element of Z5 to 0.
How is this so easy to understand when u say it, retook my uni exam 2 times now watched your video & it finally makes so much sense you need to train lecturers honestly! So good
I'm sorry that you've struggled with your exams, but I'm happy that you understand things now. Honestly, I think that many lecturers don't spend enough time on examples that illustrate the concepts. Even though the course is *abstract* algebra, the ideas grew out of looking at specific cases. It is usually easier to see the properties "at work" than trying to figure things out just from the definitions.
Your vision is beautiful 💕
Great examples. Especially φ(x)=x^2 4:39, because it's also not one-to-one!
Before (a priori, group) is dual to after (a posteriori, image) -- Immanuel Kant.
Normal subgroups are dual to homomorphism (factor groups) synthesize the kernel.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Being is dual to non-being creates becoming -- Plato.
Domains (groups) are dual to codomains (image, range).
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point, non null homotopic requires at least two points (duality) -- topology.
Polar opposites of the dyad unite into one or the monad - opposame.
"Always two there are" -- Yoda.
lots of philosophy and maths there (my favorite subjects)
@@maxpercer7119 Increasing the number of states or dimensions is an entropic process -- co-homology.
Decreasing the number of states or dimensions is a syntropic process -- homology.
Homology (convergence, syntropy) is dual to to co-homology (divergence, entropy) -- topology.
The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking.
Your mind is syntropic as it makes predictions to track targets and goals -- teleology.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line.
In a communication system the receiver of a message literally predicts the message into existence -- Shannon information theory.
There is a dual process to that of increasing entropy.
In information theory average information (entropy) becomes mutual information (syntropy) -- duality!
Y = X.
Y is an equivalent or dual description of X, all mathematical equations are dualities.
Main stream physics is currently dominated by teleophobia or eliminative materialism.
"Philosophy is dead" -- Stephen Hawking.
Teleological physics (syntropy) is dual to non teleological physics (entropy).
"Always two there are" -- Yoda.
Addition is dual to additive inverses (subtraction) -- Abstract algebra.
Multiplication is dual to multiplicative inverses (division) -- Abstract algebra.
Integration (syntropy) is dual to differentiation (entropy).
Elliptic curves are dual to modular forms.
Rational, analytic (mathematics) is dual to empirical, synthetic (physics) -- Immanuel Kant.
Noumenal (a priori) is dual to phenomenal (a posteriori) -- Immanuel Kant.
There are patterns of duality hardwired into physics, mathematics and philosophy!
Nice explanation!
Nice work .
For the final example, the ker(phi) = {1}, since it's from R*; -1 isn't in the domain.. right?
We're using R* as the real numbers except 0, so it does include negatives.
Thanks!
please do the same for z5 to z6 in another video
It would honestly be a very boring video. Assuming the group operations are addition (with respect to the appropriate mod) in each group, the only homomorphism is the "trivial" one that maps every element of Z5 to 0.