Where Exploration Leads to Academic Excellence
A Knowledge Hub by Rahul Anand
Mathematical Marvels: Insights from the Field
Even Numbers
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…
Cartesian Product
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…
Ordered Pairs
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
Tuples in Relations and their examples
Relations & Sets
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…
Even & Odd Functions
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,…
Probability Cause and Effect Problem
Question What does it mean for one event ? to cause another event ? – for example, smoking (?) to cause cancer (?)? 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…
Probability Problem: Suppose you roll a fair die two times. Let ? be the event “THE SUM OF THE THROWS EQUALS 5” and ? be the event “AT LEAST ONE OF THE THROWS IS A 4”. Solve for the probability that the sum of the throws equals 5, given that at least one of the throws is a 4. That is, solve ?(?|?).
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} =…
Calculus Castle
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Insight into the Laws of Physics
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NUMERICAL SOLUTIONS IN PHYSICS
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Atom to Atom: Insights from the Chemical Universe
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Life’s Mysteries Revealed: Exploring Biology
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