Question

If \(x y=0\) then \(x=0\) or \(y=0,\) and conversely.

Short answer

In conclusion, the statement 'if \(x y=0\), then \(x=0\) or \(y=0\)', and its converse 'if \(x=0\) or \(y=0\), then \(x y=0\)' are both true.

Step-by-step solution

Assume \(x y=0\)

We begin by assuming the given statement, that is \(x y=0\). This means that the product of \(x\) and \(y\) is equal to zero.

Demonstrate Zero Product Property

If \(x y=0\), then either \(x=0\), \(y=0\) or both. This is because any number multiplied by zero yields zero. Thus, we can now confidently say that if \(x y=0\), then \(x=0\) or \(y=0\).

Prove the converse

Now, for the converse, assume that either \(x=0\) or \(y=0\). In this case as well, multiplying \(x\) and \(y\) will yield \(0\), since any number multiplied by zero gives zero. Thus, the converse 'if \(x=0\) or \(y=0\), then \(x y=0\)' also holds true.