# Find the set ∆X

Let $X = \{0, 1, 2, 3, 4\}$, and $\Delta X = \{(x,x)~:~x\in X\}$.

Find the set $\Delta X$.

\begin{align} \Delta X=\{&(0,0), (0,1), (0,2), (0,3), (0,4),\\ & (1,0), (1,1), (1,2), (1,3), (1,4),\\ & (2,0), (2,1),(2,2), (2,3), (2,4),\\ & (3,0), (3,1), (3,2), (3,3), (3,4),\\ & (4,0), (4,1), (4,2), (4,3), (4,4)\} \end{align} Have I correctly found set $\Delta X$?

• Welcome to MSE. Please use MathJax. – José Carlos Santos Feb 17 '18 at 18:02
• No, that's $X\times{X}$. – Michael McGovern Feb 17 '18 at 18:03
• Note the difference between $\{(x,x)~:~x\in X\}$ where both elements in a pair must be the same and $\{(x,y)~:~x\in X,~y\in X\}$ where the elements in a pair could be different. – JMoravitz Feb 17 '18 at 18:05

Your set $\Delta X$ is incorrect. Note that in the definition of $\Delta X$, each member of this set is of the form $(x,x)$, so the entries in the pair must be the same. This means $\Delta X$ cannot contain elements like $(0,1)$. The answer is
$$\Delta X = \{(0,0),(1,1),(2,2),(3,3),(4,4)\}$$