Construct five different examples of sets. Describe each with the help of a proposition.

Give an example of a set which has exactly $10$ elements and express it through a defining property.

Is it possible to express every set through a defining property? Justify your answer.

If $\left\{x:p_1\left(x\right)\right\}=\left\{x:p_2\left(x\right)\right\}$, show for each $x,p_1\left(x\right)$ and $p_2\left(x\right)$ have the same truth value.

For the following word, write down the set of letters forming that word:

administration

For the following word, write down the set of letters forming the word: "misrepresentation"

For the following word, write down the set of letters forming the word: "mathematics"

For the following word, write down the set of letters forming the word: "concurrence"

For the following word, write down the set of letters forming the word: "demonstration"