LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

UNIVERSAL SET

__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 }**