Why set is not allowed duplicate value, which kind of mechanism used behind them?

Question

I am new to java, i know set is not allowed duplicate value but i don't know why set is not allowed duplicate value, Actually i am doing practically,

Declared one set and add duplicate value but no kind of error is occurring, no compile time error, no run time. why?

Answer 1

Set (Oracle Documentation)

A collection that contains no duplicate elements. More formally, sets contain no pair of elements e1 and e2 such that e1.equals(e2), and at most one null element. As implied by its name, this interface models the mathematical set abstraction.

See: http://docs.oracle.com/javase/7/docs/api/java/util/Set.html

Set (Methematics) - Quoting from wikipedia

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

Add Method

According to the documentation of the interface, if the element does not exist it is added. Otherwise, nothing changes.

boolean add(E e):

Adds the specified element to this set if it is not already present (optional operation). If this set already contains the element, the call leaves the set unchanged and returns false.

Example Implementation Code: HashSet

 /**
     * Adds the specified element to this set if it is not already present.
     * More formally, adds the specified element <tt>e</tt> to this set if
     * this set contains no element <tt>e2</tt> such that
     * <tt>(e==null&nbsp;?&nbsp;e2==null&nbsp;:&nbsp;e.equals(e2))</tt>.
     * If this set already contains the element, the call leaves the set
     * unchanged and returns <tt>false</tt>.
     *
     * @param e element to be added to this set
     * @return <tt>true</tt> if this set did not already contain the specified
     * element
     */
    public boolean add(E e) {
        return map.put(e, PRESENT)==null;
    }