Question

To determine the partitions of the set of ordered pair of integers according to the given condition.

Short answer

The collection of the subsets \(A,B\) and \(C\) is not a partition of the set \(S\).

Step-by-step solution

Given data

The set of pairs \((x,y)\) where \(x \ne 0\) and \(y \ne 0\), the set of pairs \((x,y)\) where \(x = 0\) and \(y \ne 0\), and the set of pairs \((x,y)\) where \(x \ne 0\) and \(y = 0\).

Concept used of partition of sets

The set\(S\)in a partition must be nonempty, pair wise disjoint, and have as their union.

Find partition of set

The subset according to the given conditions are,

\(A = \{ \)the set of pairs \((x,y)\), where \(x \ne 0\) and \(y \ne 0\} \)

\(B = \{ \)the set of pairs \((x,y)\), where \(x = 0\) and \(y \ne 0\} \)

\(C = \{ \)the set of pairs \((x,y)\), where \(x \ne 0\) and \(y = 0\} \)

Each ordered pair is contained in one and only one of the subsets \({A_,}\), Band C.

The three subsets are non-empty and \(S \ne A \cup B \cup C\)

But, the ordered pair \((0,0)\) does not contained in any of the defined subset.

Therefore, from the definition of partition, the collection of the subsets \(A,B\) and \(C\) is not a partition of the set \(S\).