Universal Sets
If the Set of all objects is considered under one set or belongs to one set then it is the universal set for the discussion. For example, if X, Y, Z, etc are the sets in our discussion then the set which has all the members of X,Y,Z, etc. can act as the universal set and the universal set is vary from problem to problem and it is denoted by U.
If the sets involved in a discussion are sets of some natural numbers then the set ‘S’ of all natural numbers can be taken as the universal sets and the set ‘Z’ of all whole numbers can also be taken as the universal set because all natural number is the whole number.
The definition of Universal sets implies that, in any particular problem every set is the Subset of a universal set. Since the empty set φ does not have any members, it is a subset of every other set and every set is its own subset.
Example -1
Write the set Z = { x / 3x < 20 } in the tabular form, when the universal set is natural, whole and integer number.
Ans.) As every member of Z must belong to the universal set Z, if x is the natural number and satisfying 3x < 20, x = 1, 2, 3, 4, 5, 6
So, Z = { 1, 2, 3, 4, 5, 6 }
As every member of Z must belong to the universal set as x must be a whole number satisfying 3x < 20.
So, x = 0, 1, 2, 3, 4, 5, 6
So, Z = { 0, 1, 2, 3, 4, 5, 6 }
As every member of Z must belong to the universal set, x must be an integer satisfying 3x < 20.
So, x = any negative integers, 0, 1, 2, 3, 4, 5, 6
So, Z = { ………,-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 }