Venn Diagrams | Set

Theorem 3          The total number of subsets of a finite set containing n elements is 2ⁿ.    Proof    Let A be a finite set containing n elements.Let 0 ≤ r ≤ n. Consider those subset,of A that  have r elements each.We know that the number of ways in which r elements can be chosen…

First Law for a Closed System Undergoing a Quasi-static process In many occasions it is necessary to consider a system undergoing a process rather than a cycle. The equation, ∮ dQ − ∮ dW = 0 is applicable during the system undergoing a cycle, and algebraic sum of all energies transfer across the system boundary…

Power set  Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A). That is.  P(A) = { S : S ⊂ A }. Since the empty set and the set A itself are subsets of A and are…

Intervals as subsets of R    Closed intervals  Let a and b be two given real numbers such that a < b. Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b] .   Thus,     [a, b] = {…

Function as a correspondence   Let A and B be two non-empty sets. Then a function ‘f ‘ from set A  to set B is a rule or method or correspondence which associates elements of set A to elements of set B such  that: Fig. 2(a) (i) all elements of set A are associated to…

Difference of sets Let A and B be two sets. The difference of A and B written as A – B, is the set of all those elements of A which do not belong to set B Thus A – B={ x : x ∈ A and x  ∉ B} or  A – B={ x…