Answer to Question #308303 in Discrete Mathematics for cane

Question

For each of the ff. sets, determine whether 2 is an element of that set.

a. {𝑥∈ℝ|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1}

b. {𝑥∈ℝ|𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}

c. {2,{2}}

d. {{2},{{2}}}

Answer 1

Solution (a)

The given set is {${}$ $𝑥∈ℝ|𝑥$ 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1} can be written as

$\left \{ 2, 3, 4, 5, 6, 7, 8, …\right \}$

It is clear that $2 \in \${${}$ $𝑥∈ℝ|𝑥$ 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1}

Hence 2 is an element of that set.

Solution (b)

The given set is { $𝑥∈ℝ|𝑥$ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} can be written as

$\left\{ {{1^2},\,{2^2},\,{3^2},\,....} \right\}\$

$\left\{ {1,\,\,4,\,\,9,\,\,....} \right\}\$

We can see that $2 \in \left\{ {1,\,\,4,\,\,9,\,\,....} \right\}\$

Hence $2 \notin \$ {𝑥∈ℝ|𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}

Hence 2 is not an element of that set

Solution (c)

The given set is {2,{2}} can be written as

We can see that $2 \in$ $\left \{ 2,\left \{2\right \} \right \}$

Hence 2 is an element of that set

Solution (d)

The given set is $\left \{ \left \{2\right \},\left \{2\right \} \right \}$ can be written as

We can see that $2 \notin \ \left \{ \left \{2\right \},\left \{2\right \} \right \}$

Hence 2 is not an element of that set