**Find โ ๐จ๐ โ ๐=๐ and โ ๐จ๐ โ ๐=๐ where: ๐จ๐ = {๐,๐ + ๐,๐ + ๐, โฆ } for every positive integer ๐.**

The **Answer to the Question**

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**Here's the Solution to this Question**

Let us findย $\cup_{i=1}^{\infty}A_i$ย andย $\cap_{i=1}^{\infty}A_i$ย whereย $A_i = \{i,i+1,i+2, โฆ \}$ย for every positive integerย $i$.

Taking into account thatย $A_i\supset A_j$ย forย $j>i,$ย we conclude thatย $\cup_{i=1}^{\infty}A_i=A_1=\{1,2,3,\ldots\}.$

Let us show thatย $\cap_{i=1}^{\infty}A_i=\emptyset$ย using the method by contradiction. Suppose thatย $k\in \cap_{i=1}^{\infty}A_i$ย for some positive integerย $k$. Sinceย $k\notin\{k+1,k+2,\ldots\}=A_{k+1},$ย we conclude thatย $k\notin\cap_{i=1}^{\infty}A_i.$ย This contradiction proves thatย $\cap_{i=1}^{\infty}A_i=\emptyset.$