Question

How many elements does the successor of a set with \(n\) elements have?

Sometimes the number of times that an element occurs in an unordered collection matter. Multisets are unordered collections of elements where an element can occur as a member more than once. The notationdenotes the multiset with element\({a_1}\)occurring\({m_1}\)times, element\({a_2}\)occurring\({m_2}\)times, and so on. The numbers\({m_i}\),\(i = 1,2,...,r\)are called the multiplicities of the elements\({a_i}\),\(i = 1,2,...,r.\)

Let \(P\) and \(Q\) be multisets. The union of the multisets \(P\) and \(Q\) is the multiset where the multiplicity of an element is the maximum of its multiplicities in \(P\) and \(Q\). The intersection of \(P\) and \(Q\) is the multiset where the multiplicity of an element is the minimum of its multiplicities in \(P\) and \(Q\). The difference of \(P\) and \(Q\) is the multiset where the multiplicity of an element is the multiplicity of the element in \(P\) less its multiplicity in \(Q\) unless this difference is negative, in which case the multiplicity is \(0\). The sum of \(P\) and \(Q\) is the multiset where the multiplicity of an element is the sum of multiplicities in \(P\) and \(Q\). The union, intersection, and difference of \(P\) and \(Q\) are denoted by \(P \cup Q\), \(P \cap Q\), and \(P - Q\), respectively. (Where these operations should not be confused with the analogous operations for sets). The sum of \(P\) and \(Q\) is denoted by \(P + Q\)

Short answer

The set \(A\)then contains \(n\) elements, then the successor of \(A\) contains \(n + 1\)elements.

Step-by-step solution

Step 1

Union \(A \cup B\): all elements that are either in \(A\)OR in \(B\)

The successor of\(A\)is\(A \cup \{ A\} \).

Basically, the successor of the set\(A\)is the set\(A\)with the element\(\{ A\} \)added to it.

Thus, the successor of \(A\) contains one more element than the set \(A\).

Hence, the successor of \(A\) contains \(n + 1\)elements.