Notes, summaries, assignments, exams, and problems for Mathematics

Propositional Logic

• Propositional logic: The study of arguments whose validity depends on how the statements are connected, regardless of their meaning.
• Argument: A sequence of propositions whose purpose is the implication of another proposition.
• Premise: A succession of propositions that serve as evidence.
• Conclusion: The inferred proposition.
• Proposition: A declarative sentence whose meaning can be True or False.
• Variable: An unspecified element of a reference set.
• Propositional form: An expression containing a variable that becomes a proposition when the variable is substituted by an element of a reference set.
• Simple statement: Refers to a single event.
• Tautology: A compound proposition that is True under all possible truth conditions.
• Contradiction:

Calculating Karl Pearson Correlation Coefficient

Question 11: Calculate the coefficient of correlation from the following data:

To calculate Karl Pearson's coefficient of correlation (r), we use the formula:

r = ∑dx·dy / √(∑dx² × ∑dy²)

Where:

• dx = X - X̄
• dy = Y - Ȳ

Step 1: Find the Means

First, calculate the mean of X (X̄) and Y (Ȳ):

X̄ = (12 + 9 + 8 + 10 + 11 + 13 + 7) / 7 = 70 / 7 = 10

Ȳ = (14 + 8 + 6 + 9 + 11 + 12 + 3) / 7 = 63 / 7 = 9

Step 2: Prepare the Calculation Table

Step 3: Substitute into the Formula

Substitute the sums from the table into the correlation formula:

r = 46 / √(28 × 84)
r = 46...

Introduction to Statistical Learning

Key Notation

• Y: Response (dependent/output) variable - quantitative or qualitative
• X₁,...,Xₚ: Predictors (features/covariates/independent variables)
• p: Number of predictors
• n: Number of observations
• i: Subject index (i = 1,...,n)
• j: Variable index (j = 1,...,p)
• Data: (Yᵢ, Xᵢ), i = 1,...,n

Types of Learning

• Supervised learning: Have both Y and X (prediction or inference) - PRIMARY FOCUS
• Unsupervised learning: Have X but no Y

Model for Data

Yᵢ = f(Xᵢ) + εᵢ, i = 1,...,n
Assumptions: E(εᵢ) = 0, var(εᵢ) = σ², εᵢ are mutually independent
Properties: E(Yᵢ|Xᵢ) = f(Xᵢ), var(Yᵢ|Xᵢ) = σ²

Prediction vs Inference

Floating-Point Arithmetic and Stability

Eigendecomposition: Symmetric & Positive Definite (PD)

Exponent bias: b = 011111111111 = 1023

Grampian matrix:

Maximum exponent: maxp = 1023

Minimum exponent: minp = -1023 - 51

L2 Induced Matrix Norm:

Precision range: 2**-52 <= s < 2

Note on stability: A stable algorithm cannot overcome an ill-conditioned system; it can only guarantee precision when the system is not ill-conditioned.

Special Floating-Point Values:

• np.Inf: 0111111111110
• np.NaN: 0111111111111
• np.Inf - np.Inf: NaN
• np.Inf / np.Inf: NaN
• np.Inf * 0: NaN
• np.NaN + any: NaN

Backward Error: Back-calculating from the computed result to find the equivalent perturbation of the input.

Analytical inverse and the Woodbury matrix identity are both effective...

4.1 The stages of learning as Dienes
The learning process is a process based on abstraction, generalization and communication. This process of abstraction is to accurately analyze and Dienes identifies six different stages in it:
Stage 1: introduces the individual in the middle => Game Free
2nd stage: review, manipulate, get rules => Structured Games
3rd stage: becoming aware of the common structure to games made
4th stage: representation of the common structure graphically or schematically => Stage representative
5th stage study of the properties of abstract structure, which implies the need to invent a language => Stage symbolic
6th stage: Construction of axioms and theorems => formal Stage
Her pedagogical approach

System Types
The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate or compatible system when it admits an infinite set of solutions.
Fitting and classification:

Calculating the rank of a matrix for determining

1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
• A line is proportional to another.
• A line is a linear combination of others.
Delete the third column because it is a linear...

A polygon is the portion of the plane bounded by a closed polyline. A polygon is a plane shape with straight sides. A plane is a flat surface with no thickness.
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is “closed” (all the lines connect up).

1. Circumcenter

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
It is located at the intersection of the three side bisectors. As we already know the line bisector is the locus of all the circle centers that pass through the endpoints of a segment. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that...

SPSS (Statistical Package for the Social Sciences) is a software tool used for data analysis, statistical tests, and visualization. It is widely used in research, business, and academia for handling both small and large datasets.

Understanding SPSS: Purpose and Capabilities

The primary purpose of SPSS is to simplify and streamline data management and analysis. It allows users to:

• Perform statistical calculations without complex coding.
• Generate graphs, tables, and charts for data representation.

Key Applications of SPSS Software

SPSS is applicable in multiple fields for various purposes:

• Research: Analyze survey or experimental data.
• Education: Evaluate student and staff performance.
• Healthcare: Track patient records or treatment effectiveness.
• Marketing: