If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(R=\{(x,y)\in A\times B:y=x+2\}\), what is \(R\)?

Correct answer: \(\{(1,3),(2,4)\}\)

A pair belongs to \(R\) only when its first component is in \(A\), its second component is in \(B\), and it satisfies \(y=x+2\). For \(x=1\), \(y=3\), producing \((1,3)\). For \(x=2\), \(y=4\), producing \((2,4)\). For \(x=3\), the equation gives \(y=5\), but 5 is not in \(B\), so that pair is excluded. Hence \(R=\{(1,3),(2,4)\}\), option A.