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Real Analysis Proofs

1 ♦ 0 = 0.a
0.a = 0.a

0.a = (0 + 0).a [Since 0 is the additive identity]
0.a = 0.a + 0.a [Distributive Axiom]

-0.a + 0.a = -0.a + 0.a + 0.a [Adding (-0.a) to both sides, the additive inverse of 0.a]
0 = (-0.a + 0.a) + 0.a [Associative Axiom]

0 = 0 + 0.a

0 = 0.a [Since 0 is the additive identity] ♠

2 ♦ -a = -1(a)
(-a) = (-a)

(-a) = (-a) + 0 [Additive Identity]

(-a) = (-a) + 0.a [Since 0 = 0.a (By above theorem (1))]
(-a) = (-a) + (1 + (-1))a [Additive Inverse of 1]

(-a) = (-a) + (1.a + (-1)a) [Distributive Axiom]
(-a) = (-a) + (a + (-1)a) [Multiplicative Identity]
(-a) = ((-a) + a) + (-1)a [Associative Axiom]
(-a) = 0 + (-1)a [Additive Identity]

(-a) = (-1)a [Additive Identity] ♠

3 ♦ -(a + b) = (-a) + (-b)
-(a + b) = -(a + b)

–(a + b) = (-1)(a + b) [By above theorem (2)]
(-1)(a + b) = (-1)(a) + (-1)(b) [Distributive Axiom]

-(a + b) = (-a) + (-b) [By above theorem (2)]

♦ Sup(A + B) = Sup(A) + Sup(B)
∀ a ∈ A and ∀ b ∈ B;
a ≤ Sup(A) and b ≤ Sup(B)

∴ a + b ≤ Sup(A) + S…

Python Laboratory Excersices

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L1_2.2 Python Programs
Step 1: Type the Python program using a text editor such as gedit or vi or emacs or Kwrite and save as test1.py ($ gedit test1.py)
Step 3: Modify the above program to display the source code.

# Step 1# ******** ## display Hello Worldprint"Hello world !\n"# Step 3# ******** #print"# display Hello World\n"print"print \"Hello world !\\n\"\n"

Functions

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Definition
Let A and B be two subsets of ℝ
A relation from A to B is a rule which assigns a member of B to each member of A.
Definition
Let A and B are two subsets of ℝ
A function from A to B is a rule which assigns a unique member of B of each number of A.

About Padmal's Blog

This is my very first blog. Learning and implementing about blogging and discovering it's trends, basically this blog will help anyone out with Mathematics, Python Programming Fundamentals at the university level.
Mathematics will cover Logic and Set Theory [Propositions, Quantifiers, Techniques of Proof, Sets, Relations, Functions], Real Analysis [Intervals, Supremum and Infemum, Completeness Axiom, Limit of a Function, Differentiability, Rolle’s Theorem, L’ Hospital’s Rule, Sequences, Convergence of Series: Ratio Test, Comparisons Test, Limit Comparison Test, Maclaurine’s Integral Test; Taylor’s Series, Integration of Continuous Functions, Definite and Indefinite Integrals], Complex Numbers, Vectors, and Matrices [Algebra of Complex Numbers, De Moriver’s Theorem, Argand Diagram, Vector Algebra, Scalar Triple Product, Vector Triple Product, Line and Plane, Matrix Algebra, Echelon Forms, Rank, Determinants, Eigen Values, Spectral Radius, Matrix Norms: Maximum Row Sum, Maximum Co…

Intervals

Defenition
Let A ≤ R; A is said to be an interval if following condition is satisfied.

If a, b ∈ A with a < b, then
a < x < b ⇒ x ∈ A

Eg: A = {1, 2, 3, 4}
2, 3 ∈ A & 2 < 2.5 < 3 & 2.5 ∉ A

Therefore A is not an interval

Inequalities

Definition
ℝ+ = {a | a ∈ ℝ and a is located to the R.H.S. of 0 in the number line}
ℝ- = {a | a ∈ ℝ and a is located to the L.H.S. of 0 in the number line}
{0} = The single point 0
Then, ℝ = ℝ- ∪ {0} ∪ ℝ+

Irrational Numbers

In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable(and the rationals countable) it follows that almost all real numbers are irrational.
Note: ∃ a (There is...)∃ a1, a2, a3 ... (There are...)