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In physics and geometry, the concept of "three dimensions" refers to a spatial framework necessary to describe the position or location of an object fully. Each dimension provides a unique axis that, together with the others, can describe any point in space. Here’s a...
Scientific notation is a way to express very large or very small numbers in a compact form. It's especially useful in fields like science, engineering, and mathematics where such numbers frequently occur. The notation is based on powers of 10. Here's the general form:...
Find \( \lim_{x \to 3} (2x + 5) \) Solution:To solve this limit, we substitute the value of \( x \) directly because the function is continuous at \( x = 3 \).\( \lim_{x \to 3} (2x + 5) = 2(3) + 5 = 6 + 5 = 11 \)
5 Basic Problems on Limits just to refresh your mind. Problem 1 Find the limit: \( \lim_{x \to 2} (3x - 4) \) Solution: To solve this limit, we substitute the value of \(x\) directly because the function is continuous at \(x = 2\). \[ \lim_{x \to 2} (3x - 4) = 3(2) -...
The multiplication rule in probability is used to find the probability of the intersection of two or more independent event
Finding the Expected Value μ (mean) of Random Variable that has Poisson Distribution
In probability theory and statistics, λ (lambda) is a parameter used to represent the average rate or average number of events occurring in a fixed interval in the context of a Poisson distribution.
Derive the formula of Variance of the Poisson Distribution
uniformly distributed aerosol particles between 2 and 6 nanometers
Probability theory is a fascinating subject that has many applications in the real world. Understanding the basics of random variables and probability distributions is essential for anyone working in a field that deals with uncertainty. By mastering probability theory, you can make better decisions and improve your ability to analyze and interpret data.
Practical illustrations of Random Variables that we are exposed to in our daily life
The six trigonometric functions are defined below. Refer to the above diagram to get the relational picture. sinθ = \( \dfrac {\mathrm{perpendicular}} {\mathrm{hypotenuse}} = \dfrac {p}{h} \) cosθ = \( \dfrac {\mathrm{base}} {\mathrm{hypotenuse}} = \dfrac {b}{h}...
Pythagoras’ theorem is stated as : The sum of the areas of the two squares on the perpendicular(p) and base(b) of a right-angle triangle is equal to the area of the square on the hypotenuse(h). i.e. p2 + b2 = h2
A right-angle triangle is a triangle in which one of the angles measures 90°. Right-angled triangles have wide applications in mathematics and physics and as such, it became convenient to have specific names for their sides so that the problem statement in mathematics...
The trigonometric functions can be described on an x-y coordinate plane (Euclidean plane) using a circle of radius 1 unit and cutting a sector that subtends an angle θ at the centre. Refer to the diagram below for details.
By definition, L (length of arc) = ( \dfrac { \mathrm{θ_{deg} } } {360} ) × Circumference (arc length is proportional to angle, one complete arc subtends 360° at center) Also, Circumference = 2 𝛑 r Hence, L = ( \dfrac { \mathrm{θ_{deg} } } {360} ) × 2 𝛑 r ...
A number not divisible by 2 is called an odd number. Any number whose unit digit(last digit) is either 1,3,5,7 or 9 is an odd number. The set of all odd numbers is represented as Odd numbers = { 2n+1: n ∈ Z } where Z is the set of all integers. When an even...
A number divisible by 2 is called an even number. All numbers whose unit digit(last digit) is either 0,2,4,6 or 8 is an even numbers. The set of all even numbers is represented as follows: Even numbers = { 2n: n ∈ Z } where Z is the set of all integers. Zero is...
The cartesian product of two sets A and B is defined as a set formed by all the possible ordered pairs of elements from A and B, such that the first element comes from set A and the second element comes from set B. The cartesian product is denoted as A × B. Let a and...
An ordered pair is a 2-tuple formed by taking two elements (generally numbers but can be alphabets, characters, words or symbols). The general form of representation is (a, b) where a and b represent two distinct objects. The important thing with ordered pairs is that...
Tuples in Relations and their examples
A Relation from set A to set B is defined as a set of ordered pairs formed from the elements of set A and B. In other words, a relation is a subset of the cartesian product of sets A and B. The subset is derived by establishing predicate filter(s) or criteria stating...
A function is said to be an even function if the sign of the image does not change when the sign of the preimage changes. Conversely, a function is called an odd function when the sign of the image changes when the sign of the preimage changes. For Even functions,...
Question What does it mean for one event 𝐶 to cause another event 𝐸 - for example, smoking (𝐶) to cause cancer (𝐸)? There is a long history in philosophy, statistics, and the sciences of trying to clearly analyze the concept of a cause. One tradition says that causes...
Solution We have A = (1,4), (2,3), (3,2), (4,1) B = (1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (4,1), (4,2), (4,3), (4,5), (4,6) \( P(A|B) = \dfrac {P(A∩B)}{P(B)} \) \( A∩B = (1,4), (4,1) \) The sample space comprises of 6×6 = 36 eventsHence,\( P(A∩B) = \dfrac{2}{36} =...
We have \( \lim_{\theta\to0} { \sin\theta \over \theta } \) = 1 Consider the below diagram. We have r = radius of the circle.A = centre of the circle.The sector ⌔ formed by the arc BD subtends an angle θ at the centre. Case 1 : θ > 0 i.e. θ is +ve Let 0 ≤ θ ≤ \(...
To prove : lim\( _{x \to a} { x^n - a^n \over x - a } = na^{n-1} \) where n is a rational number Proof: Let \( x = a + h \) Then as \(x \to a \), we have \(h \to 0 \) Now, \( \lim_{x \to a} { x^n - a^n \over x - a } = \lim_{h \to 0} { (a...
Proof : We have, lim\(_{θ\to 0} { \dfrac {\mathrm tan \mathrm θ}{ \mathrm θ} } \) = lim\(_{θ\to 0} { \dfrac {\mathrm \sin \mathrm θ} {\mathrm θ \mathrm \cos\mathrm θ} } \) \( \{∵ \tan\theta = \dfrac...
As θ → 0, we have cosθ → 1 Proof : When θ = 0, We have, lim\(_{θ\to 0} \cos \)θ = cos0 = 1 { ∵ cos0 = 1 } Hence, lim\(_{θ\to 0} \cos \)θ = 1
Derivative of \({e}^x\) using the First Principle Let \(y\) = \({e}^x\)∴ \(y + δy\) = \({e}^{x + δx}\)∴ \(δy\) = \({e}^{x + δx}\) - \({e}^x\)or \(δy\) = \({e}^{x}\) . \( [ {e}^{δx} - 1 ]\)Dividing each side by δx </h3>or \(\dfrac {δy}{δx}\) = \( \dfrac { {e}^{x}...
Derivative of \( sinθ \) using the First Principle Let \(y\) = \( sinθ \) ∴ \(y + δy\) = \( sin(θ + δθ) \) ∴ \(δy\) = \( sin(θ + δθ) \) - \( sinθ \)From Trigonometry , we have \( sin(A-B) \) = 2.\( sin \dfrac {(A-B)}{2} \).\( cos \dfrac {(A+B)}{2} \)Using the above...
Derivative of \( cosθ \) using the First Principle Let \(y\) = \( cosθ \) ∴ \(y + δy\) = \( cos(θ + δθ) \) ∴ \(δy\) = \( cos(θ + δθ) \) - \( cosθ \)From Trigonometry , we have \( cos(A-B) \) = -2.\( sin \dfrac {(A+B)}{2} \).\( sin \dfrac {(A-B)}{2} \)Using the above...
De Morgan’s First Law The complement of the union of two sets is equal to the intersection of their complements i.e. (A ∪ B )' = A' ∩ B' De Morgan’s Second Law The complement of the intersection of two sets is equal to the union of their complements i.e. (A ∩ B )’ =...
A cartesian product between two sets is defined as the set consisting of all possible ordered pairs that can be formed by taking one element from each of the sets at a given time. If A and B are two sets such that a ∈ A and b ∈ B, then the cartesian product between A...
In mathematics, a tuple or a sequence is a list of objects arranged in an order. Such a list may have repeated objects but the order is more important. Such sequences or tuples are denoted as ( t1, t2, t3, …., tn ) where tn is the nth element of the list. Below are a...
An ordered pair is a 2-tuple formed by taking two elements (generally numbers but can be alphabets, characters, words, or symbols). The general form of representation is (a, b) where a and b represent two distinct objects. The important thing with ordered pairs is...
The cartesian product of two sets A and B is defined as a set formed by all the possible ordered pairs of elements from A and B, such that the first element comes from set A and the second element comes from set B. The cartesian product is denoted as A × B....
The intersection of two sets A & B is defined as a set that contains only those members which are common to both A and B. The intersection operation is denoted by the symbol ∩. Remember, for two disjoint sets (sets having no common elements), the...
The union of two sets A & B is defined as a set that contains all the member elements of A and B. the union operation is denoted by the symbol ∪. One point to remember here is that the union of two or more sets always gives a set with distinct...
The complement of a set A (that is a subset of a universal set U) is defined as a set that contains all the member elements and all subsets of U that are not part of the set A. The complement of a set is denoted using the symbol ‘ or c. Hence, A’ =...
Venn diagrams are the pictorial or graphical representation of sets and the various relationships that exist between sets. The representation consists of a rectangular box representing the universal set(U). All sets that are in context are drawn as circles...
When working with sets, a reference superset that contains all the sets and their subsets in context (i.e. the sets which are currently under consideration) is called a universal set. It provides a base set from which all of the sets of interest can be...
Partitioning of a set is distributing the member elements of a set among a group of non-empty subsets in such a way that each member lies in only one of these subsets. ⇒ ∅ ( Empty Set ) cannot be the partition of any set. Examples ⇒ The set { 1, 2, 3 } can be...
Two sets A and B are called disjoint when they have no element in common (except the empty set ∅ }.
The set of all possible subsets of a set S is called the power set of S, written as P(S). Examples ⇒ The power set of { ‘a’ } is { ∅, {‘a’} }The power set of { 2, 3 } is { ∅, {2}, {3}, {2,3} }The power set of {1, 2, 3 } is { ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},...
If A is a subset of B but A ≠ B, then A is called the proper subset of B, and B is called the proper superset of A. This relationship is represented as below A ⊂ B ( A is a proper subset of A )B ⊃ A ( B is a proper superset of A ) Examples ⇒ {...
If there are two sets A & B such that every element of A is also in B, then A is called a subset of B. In other words, A is contained in B. B is called the superset of A. In the set theory, this relationship is depicted as below...
Two sets A and B are said to be equivalent(≡) if each element of A is also an element of B and each element of B is also an element of A. If elements are repetitive in one set, then it is not required for it to repeat in the other set for the two sets to be...
Two sets A and B are said to be equal(=) if they have the same elements. The elements may not be in the same order. If an element appears n times in one set, then it must also appear n times in the other set. Mathematically, two sets A and B are equal if For each...
A set that has a definite number of elements is called a finite set else it is called an Infinite set. ⇒ A null set is a finite set. ⇒ For finite set S, n(S) is a finite number. ⇒ The standard mathematical sets like N, Z, R, etc. are all...
A set that has exactly one member is called a singleton set. { 1 } , { ‘a’ } , { x3 | x ∈ N , 2 < x < 3 } are all singleton sets.
A set that does not have any members is called an empty set. ⇒ Such sets are represented as {} or using the symbol ∅ (derived from Φ) which has been explicitly developed to designate an empty set. ⇒ Sometimes { ∅ } is also...
The number of elements in a set is called cardinality. The cardinality of a set A is generally represented by |A| or n(A) meaning the number of elements in set A. Examples: The cardinality of the set { 1, 45, 2, 34 } is 4. The cardinality of...
The position of elements in a set does not change the value or the meaning of the set. The above statement signifies that { 1, 2, 3 } and { 1, 3, 2 } and { 2, 3, 1 } are all the same set. A set is primarily a collection and not a sequential representation of elements....
As defined earlier, a set is a collection or group of objects. These objects are called members of the Set. This relationship is represented by using the symbol ∈. The symbol ‘∈’ means “is a member of ” or “belongs to” or “is an element of”. The...
In mathematical terms, the members of sets are called elements. A set is represented by enlisting its member elements within curly brackets. There are three general conventions adopted for set representation – Roster, Ellipsis & Set-Builder form. Roster form...
What is a set? A Set is a collection of items. The collection can be either real-world objects or imaginary or theoretical entities. It can be a collection of numbers, alphabets, colors, countries’ names, etc.