Question #308303

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



a. {๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ ๐‘”๐‘Ÿ๐‘’๐‘Ž๐‘ก๐‘’๐‘Ÿ ๐‘กโ„Ž๐‘Ž๐‘› 1}



b. {๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘กโ„Ž๐‘’ ๐‘ ๐‘ž๐‘ข๐‘Ž๐‘Ÿ๐‘’ ๐‘œ๐‘“ ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ}



c. {2,{2}}



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

1
Expert's answer
2022-03-09T15:31:05-0500

Solution (a)


The given set is {{} ๐‘ฅโˆˆRโˆฃ๐‘ฅ๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ ๐‘”๐‘Ÿ๐‘’๐‘Ž๐‘ก๐‘’๐‘Ÿ ๐‘กโ„Ž๐‘Ž๐‘› 1} can be written as


{2,3,4,5,6,7,8,โ€ฆ}\left \{ 2, 3, 4, 5, 6, 7, 8, โ€ฆ\right \}


It is clear that 2 \in \{{} ๐‘ฅโˆˆRโˆฃ๐‘ฅ๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ ๐‘”๐‘Ÿ๐‘’๐‘Ž๐‘ก๐‘’๐‘Ÿ ๐‘กโ„Ž๐‘Ž๐‘› 1}


Hence 2 is an element of that set.




Solution (b)


The given set is { ๐‘ฅโˆˆRโˆฃ๐‘ฅ๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘กโ„Ž๐‘’ ๐‘ ๐‘ž๐‘ข๐‘Ž๐‘Ÿ๐‘’ ๐‘œ๐‘“ ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ} 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 \ {๐‘ฅโˆˆโ„|๐‘ฅ ๐‘–๐‘  ๐‘กโ„Ž๐‘’ ๐‘ ๐‘ž๐‘ข๐‘Ž๐‘Ÿ๐‘’ ๐‘œ๐‘“ ๐‘Ž๐‘› ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ}


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 2โˆˆ2 \in {2,{2}}\left \{ 2,\left \{2\right \} \right \}


Hence 2 is an element of that set



Solution (d)


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


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


Hence 2 is not an element of that set



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