Proposition. Suppose that $\forall a,\exists a^{-1}\textrm{ such that }a^{-1}a=1$, then $\forall a,aa^{-1}=1$, where $a^{-1}$ is the same as that in the hypothesis.

Proof. Let $a$ be arbitrary.
$$\begin{align*}a^{-1}a&=1\\ aa^{-1}a&=a\\ aa^{-1}aa^{-1}&=aa^{-1}\\ (aa^{-1})^{-1}aa^{-1}aa^{-1}&=(aa^{-1})^{-1}aa^{-1}\\ aa^{-1}&=1.\quad\square\end{align*}$$

It’s impossible to prove this proposition if we delete $\forall a$, because just $a^{-1}a=1$ is too weak and has nothing to do with the value of $aa^{-1}$.

On page 90 of How to Think about Abstract Algebra, Lara Alcock says,

An element has an inverse if and only if every element appears in its row [in the multiplication table of a binary operation].

Let’s prove this. ($\Rightarrow$) Suppose $gg^{-1}=1$ and $a$ is an arbitrary element. We want to prove that $\exists h$ such that $gh=a$. Just pick $h=g^{-1}a$.

($\Leftarrow$) If $g$ is not invertible, then $1$ does not appear in its row.

s = 'print("s =", repr(s))\nprint(s)'
print("s =", repr(s))
print(s)

Green and Tao proved that there are arbitrarily long arithmetic progressions in primes. How about infinitely long progressions? The answer is negative but I don’t know how to prove it!

MSc in Data Science and Machine Learning

• Core Modules
  • • DSA5101 Introduction to Big Data for Industry
    • DSA4212 Optimization for Large-Scale Data-Driven Inference
    • DSA5102X Foundations of Machine Learning
    • CS5224 Cloud Computing
    • DSA5201 DSML Industry Consulting and Applications Project
• Elective Modules
  • Graduate Certificate in Deep Learning for Data Scientists
    • DSA5202 Advanced Topics in Machine Learning
    • DSA5204 Deep Learning and Applications
    • CS5242 Neural Networks and Deep Learning
  • Graduate Certificate in Data Mining for Industry
    • CS5228 Knowledge Discovery and Data Mining
    • ST5227 Applied Data Mining
  • Graduate Certificate in Big Data for Industry
    • CS5344 Big-Data Analytics Technology
    • ST5201 Statistical Foundations of Data Science
  • Graduate Certificate in Data Science for Computer Vision
    • CS4243 Computer Vision and Pattern Recognition
    • CS5240 Theoretical Foundation of Multimedia
    • DSA5203 Visual Data Processing and Interpretation
  • Graduate Certificate in Data Science for Quantitative Finance
    • QF5204 Numerical Methods in Quantitative Finance
    • DSA5205 Data Science in Quantitative Finance
    • ST5202 Applied Regression Analysis
  • Graduate Certificate in Health Informatics
    • SPH5414 Informatics for Health
    • SPH5104 Analytics for Better Health
    • SPH5411 Information Technology in Healthcare
  • Cluster in Mathematics
    • MA4230 Matrix Computation
    • MA5232 Modelling and Numerical Simulations
    • MA5266 Optimization
  • Cluster in Statistics
    • ST5207 Nonparametric Regression
    • ST5210 Multivariate Data Analysis
    • ST5225 Statistical Analysis of Networks
  • Cluster in Computing
    • CS4248 Natural Language Processing
    • CS5234 Algorithms at Scale
    • CS5246 Text Mining
    • CS5425 Big Data Systems for Data Science
    • IS5152 Data Driven Decision Making

Master of Computing, Artificial Intelligence Specialization

• Core Modules
  • CS5446 AI Planning and Decision Making
  • Choose one:
    • CS5339 Theory and Algorithms for Machine Learning
    • CS5242 Neural Networks and Deep Learning
    • IS5152 Data-Driven Decision Making
  • CS5340 Uncertainty Modelling in AI
• Elective Modules
  • CS4243 Computer Vision and Pattern Recognition
  • CS4244 Knowledge Representation and Reasoning
  • CS4248 Natural Language Processing
  • CS5215 Constraint Programming
  • CS5228 Knowledge Discovery and Data Mining
  • CS5242 Neural Networks and Deep Learning
  • CS5246 Text Mining
  • CS5260 Neural Networks and Deep Learning II
  • CS5469 Fundamentals of Logic in Computer Science
  • CS5477 3D Computer Vision
  • CS5478 Intelligent Robots: Algorithms and Systems
  • CS5339 Theory and Algorithms for Machine Learning
  • CS5461 Algorithmic Mechanism Design
  • CS6207 Advanced Natural Language Processing
  • CS6208 Advanced Topics in Artificial Intelligence
  • CS6216 Advanced Topics in Machine Learning
  • CS6244 Robot Motion Planning and Control
  • IS5152 Data-Driven Decision Making
  • IS5006 Intelligent System Deployment
  • IS4242 Intelligent Systems – Tools, Techniques, and Applications
  • BT4014 Analytics Driven Design of Adaptive Systems
  • BT4240 Machine Learning for Predictive Data Analytics