A universal set is a set which contains all the elements or objects of other sets,
including its own elements. It is usually denoted by the symbol ‘U’.
Example: Let us say, there are three sets named as A, B and C. The elements of all sets A, B and C is defined as;
A={1,3,6,8}
B={2,3,4,5}
C={5,8,9}
Find the universal set for all the three sets A, B and C.
Answer:
By the definition we know, the universal set includes all the elements of the given sets.
Therefore, Universal set for sets A, B and C will be,
U={1,2,3,4,5,6,8,9}
If Universal set contains Sets A, B and C, then these sets are also called subsets of Universal set. Denoted by;
Note :
It is often convenient to use the symbol ‘⇒’ which means implies.
Using this symbol, we can write the definition of subset as follows:
A ⊂ B if a ∈A ⇒ a ∈ B, where A, B are two sets.
We read the above statement as “A is a subset of B if 'a' is an element of A implies
that 'a' is also an element of B”.
Since the empty set φ has no elements, we consider that φ is a subset of every set.
If A is not a subset of B (A ⊄ B), that means there is at least one element in A that
is not a member of B