MATHEMATICS
Resources & Insights

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,\( P(A∩B) = \dfrac{2}{36} = \dfrac{1}{18} \)\( P(B) = \dfrac{11}{36} \) Thus, \( P(A|B) = \dfrac {\dfrac{2}{36} } { \dfrac{11}{36} } \) or \( P(A|B) = \dfrac{2}{11} \) (Required probability) READ MORE...

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’ ∪ B’ READ MORE...

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 and B is denoted as A × B and is evaluated as { (a,b) } where a ∈ A and b ∈ B. Let A = { 1,2 } and B = { x,y } The cartesian product of A and B denoted as A × B = { (1,x) , (1,y), (2,x), (2,y) } The cartesian product of B and A, denoted as […] READ MORE...

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 few examples of tuples. (x, y) (1, 2, 3, 4, 5) (1, 4, 9, 16) (a, e, f, j, k) (red, blue, green) (+, -, *, ×, ÷, ?) Please note ☞ A tuple with n elements is called an n-tuple. A tuple is written by enlisting the elements in the required order and enclosed within parentheses. The elements are separated by a comma. (1, 5, 9, 2, 3, 34) […] READ MORE...

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 the ordering of the participating elements is important i.e. (a, b) is different from (b, a) unless a=b (a,b) ≠ (b,a) unless a=b Examples of Ordered pairs : (1,2)(a,b)(-172,45.98)(x,3) Remember ☞ (a, b) ≠ (b, a), unless a = b.  If (a1, b2) = (a2, b2) ⇒ a1=a2 & b1=b2 Ordered pairs are widely used in set theory, calculus, relations, and function theories and in the representation of intervals for functions on numbers lines, and axis, for laws and […] READ MORE...

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 b represent arbitrary elements from set A and set B respectively. So, a ∈ A and b ∈ B. Then A × B = { (a,b) | a ∈ A, b ∈ B }. Remember ☞ ​If A has m elements and B has n elements then their cartesian product will have m × n elements. If n(A) = m and n(B)=n ,  then n(A × B) […] READ MORE...

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 intersection results in an empty set (∅). Example ⇒If A = { 1, 2, 3, 4 } and B = { 1, 4, 5, 6 }Then A ∩ B = { 1, 4} Similarly, if A = { 1, 1, 1, 2, 3 } and B = { 1, 1, 2, 2, 4 }Then A ∩ B = { 1, 2 } Similarly, if A = { dog, cat, cow } and B = { peacock, bull […] READ MORE...