MATHEMATICS
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Tuples in Relations and their examples READ MORE...

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 condition that evaluates the qualifying ordered pairs from the cartesian product to be included in the subset as specified by the relation. A relation is uni-directional i.e. if a relation exists from A to B then it does not imply that a relation exists from B to A as well. Also, (a,b) ≠ (b,a) Let a ∈ A and b ∈ B. Let (a,b) be an […] READ MORE...

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, f(x) = f(-x). For the Odd function, f(x) = -f(-x). Examples of Even Function: f(x) = x2 .We have f(1) = 1 and f(-1) = 1 hence f(1) = f(-1).This is true fora ∀ x ∈ R.Another example would be mod function |x|. Examples of Odd Function: f(x) = x3 .We have f(1) = 1 and f(-1) = -1 hence f(1) = -f(-1).This is true for ∀ x ∈ R. READ MORE...

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 raise the probability of their effects; we may write this symbolically as \( ?(?|?) > ?(?)  \) -  -  -  -  -  -  -  -  -  - (1) a) Does equation (1) imply that ?(?|?) > ?(?)? If so, prove it. If not, give a counter-example. b) Another way to formulate a probabilistic theory of causation is to say that \( P (E | C) > P(E | C^c)    \)  […] READ MORE...

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...