Question

Find the successors of the following sets.

1. \(\{ 1,2,3\} \)
2. \(\emptyset \)
3. \(\{ \emptyset \} \)
4. \(\{ \emptyset ,\{ \emptyset \} \} \)

Short answer

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 elements\(\{ A\} \)added to it.

\(\emptyset \) represents the empty set and the empty set does not contain any elements.

Step-by-step solution

Step 1

a) The successor of \(\{ 1,2,3\} \) is \(\{ 1,2,3\} \cup \{ \{ 1,2,3\} \} \)

The unions contains all elements in either of the two sets:

\(\{ 1,2,3,\{ 1,2,3\} \} \)

Step 2

b) The successor of \(\emptyset \)is \(\emptyset \cup \{ \emptyset \} \)

The unions contain all elements in either of the two sets. Note that the empty set does not contain any elements

\(\{ \emptyset \} \)

Step 3

c) The successor of \(\emptyset \)is \(\{ \emptyset \} \cup \{ \{ \emptyset \} \} \)

The unions contain all elements in either of the two sets:

\(\{ \emptyset ,\{ \emptyset \} \} \)

Step 4

d) The successor of \(\{ \emptyset ,\{ \emptyset \} \} \)is \(\{ \emptyset ,\{ \emptyset \} \} \cup \{ \{ \emptyset ,\{ \emptyset \} \} \} \)

The unions contain all elements in either of the two sets:

\(\{ \emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} \} \)

Thus, we conclude that:

1. \(\{ 1,2,3,\{ 1,2,3\} \} \)
2. \(\{ \emptyset \} \)
3. \(\{ \emptyset ,\{ \emptyset \} \} \)
4. \(\{ \emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} \} \)