Set Theory : Equal and Equivalent Sets

Definition & Examples:        

Equal Sets 

The two given sets have member of elements and the number of elements are exactly same, then the sets are called Equal Sets. 

The elements could be anything like picture, numbers, alphabets. 

The repetition of elements and the order of elements does not have any relevance.

In other way we can say that,

 Two sets are Equal Sets, if they have exact same elements.

Example :(1)  A = {1, 2, 3, 4}

                    B = {4, 2, 1, 3}

The order in which the elements of a set are written is not important.

(2)    A = {X : X is positive integer and 3x ≤ 9}

         B = {X : X is positive integer and x ≤ 3}

Therefore, 
                          A = 3x ≤ 9
                          A = x ≤ 3
                          A = {1, 2, 3,}

                    and B = {1, 2, 3,}

Here, set A and set B are equal sets, in both examples because the elements of A and B are same.

(3)                A = {a, b, c, d}

                    B = {d, c, a, b}

                    C = {a, a, b, b, c, c, d, d}

Here all the three sets set A, set B and set C are equal sets because their elements are same irrelevance of order and repetition.

The two sets A and B are called equal sets if all the elements of A are present in B and all the elements of B are present in A, or we can say that if both sets A and B are the subsets of each other then that are equal sets.

A ⊂ B and B ⊂ A ⇔ A = B

A is subset of B and B is subset of A, it means that A is equal to B. 

Example:          X = {t, a, b, l, e}

                       Y = {a, t, b, e, l}

                       Z = {t, a, b,}

Here all elements of set X and set Y are same, or we can say that  X is the subset of Y and Y is the subset of X, so X = Y.

But when this condition does not fulfill, or X is not subset of Y and Y is not the subset of X then the sets are called unequal sets.

In above example

Z ⊂ X and X ⊄ Z ⇔ X ≠ Z

it means all the elements of Z are present in X but all the elements of X are not present in Z, so X is not equal to Z or X and Z are unequal sets.

Cardinality of a equal set

The cardinality of a set X is n(X) = a, where a represents the number of elements of set X.

In above example, the cardinality of the set X is 5 and set Y is also 5.

n(X) = 5
n(X) = n(Y) = 5

So, two sets will equal if the number of elements or cardinal terms  are equal.

n(X) = n(X) 

Equal sets are denoted by the symbol “=” i.e. equality.

Unequal sets are denoted by the symbol “≠” i.e. not equal to.

In above example set X = set Y, set X equal to set Y.

But set X ≠ set Z, set X is not equal to set Z.

                      Equivalent Sets 

Two sets are Equivalent if they have different elements but have the same number of elements.
or number of elements are same. The elements need not to be same. 

 Example :(1)  A = {1, 2, 3, 4}

                      B = {a, b, c, d}

 (2)  A = {Sunday, Monday, Saturday}

       B = {may, march, July}

since, the two sets A and B contain the same number of elements therefore they are equivalent sets. 

Equivalent sets are denoted by the symbol “~” or ≣ i.e. equivalent.

Sets are said to be equivalent if the cardinality of sets are same.

All empty sets has zero cardinality, so all the empty sets are equivalent sets.