Equal And Equivalent Sets

Two sets can either be equivalent, equal or unequal to each other. Let us look at them one by one.

Equal Sets

Two sets A and B can be equal only if each element of set A is also the element of the set B. Also, if two sets are the subsets of each other, they are said to be equal. This is represented by:

\(A\) = \(B\)

\(A~⊂~B\) and \(B~⊂~A\)\(A\) = \(B\)

If the condition discussed above is not met, then the sets are said to be unequal. This is represented by

\(A\)\(B\)

For e.g. If \(P\) = {\(1,3,9,5,-7\)} and \(Q\) = {\(5,-7,5,3,1,9,-7\)}, then \(P\) = \(Q\). This is because no matter how many times an element is repeated in the set, it is only counted once. Also, order doesn’t matter for the elements in a set. So, to rephrase in terms of cardinal number, we can say that

If \(A\) = \(B\), then \(n(A)\) = \(n(B)\) and for any \(x~∈~A\), \(x~∈~ B\) too.

Let us now go ahead and find when the given two sets are equivalent.

Equivalent Sets:

To be equivalent, the sets should have the same cardinality. This means that there should be one to one correspondence between elements of both the sets. Here, one to one correspondence means that for each element in the set \(A\), there exists an element in the set \(B\) till the sets get exhausted.

Definition 1: If two sets \(A\) and \(B\) have same cardinality, if there exists a objective function from set \(A\) to \(B\).

Definition 2: Two sets \(A\) and \(B\) are said to be equivalent if they have the same cardinality i.e. \(n(A)\) = \(n(B)\).

Examples of Equal And Equivalent Sets

If \(P\) = {\(1,-7,200011000,55\)} and \(Q\) = {\(1,2,3,4\)}, then \(P\) is equivalent to \(Q\).

If \(C\) = {\(x~:~x~ is ~an~ integer\)} and \(D\) = {\(d~:~d ~is~ a~ natural~ number\)}, then \(C~ is~ equivalent~ to~ D\).

Some important points:

  • All the null sets are equivalent to each other.
  • If \(A\) and \(B\) are two sets such that \(A\) = \(B\), then \(A\) is equivalent to \(B\)..This means that two equal sets will always be equivalent but the converse of the same may or may not be true.
  • Not all infinite sets are equivalent to each other. For e.g. the set of all real numbers and the set of integers.

Practise This Question

(AB)(BC)=___