Question

Va rog frumos sa ma ajutati!Ofer 100 de puncte:)

Answer 1

$E=\left \{ m\in \mathbb{N}\:| \: {2}^{m} < {3}^{2} \right \}$

$F=\left \{ n\in \mathbb{{N}^{*} }\:| \: n = m - 1,m \: \in \: E\right \}$

$a) {2}^{m} < {3}^{2} = > {2}^{m} < 9$

$pt. \: m = 0 = > {2}^{0} = 1 < 9$

$pt. \: m = 1 = > {2}^{1} = 2 < 9$

$pt. \: m = 2 = > {2}^{2} = 4 < 9$

$pt. \: m = 3 = > {2}^{3} = 8 < 9$

$= > m \: \in \: \left \{ 0,1,2,3 \right \}$

$= > E=\left \{ 0,1,2,3 \right \}$

$n = m - 1$

$pt. \: m = 0 = > n = 0 - 1 = - 1 \: \notin \: \mathbb{{N}^{*}}$

$pt. \: m = 1 = > n = 1 - 1 = 0 \: \notin \: \mathbb{{N}^{*}}$

$pt. \: m = 2 = > n = 2 - 1 = 1 \: \in \: \mathbb{{N}^{*}}$

$pt. \: m = 3 = > n = 3 - 1 = 2 \: \in \: \mathbb{{N}^{*}}$

$= > m \: \in \: \left \{ 2,3 \right \}$

$= > F=\left \{ 2,3 \right \}$

$b)E\cup F=\left \{ 0,1,2,3 \right \}\cup\left\{2,3\right\}=\left \{ 0,1,2,3 \right \}$

$E\cap F=\left \{ 0,1,2,3 \right \}\cap\left\{2,3\right\}=\left \{ 2,3 \right \}$

$E\setminus F=\left \{ 0,1,2,3 \right \}\setminus\left\{2,3\right\}=\left \{ 0,1 \right \}$

$F\setminus E=\left \{ 2,3 \right \}\setminus\left\{0,1,2,3\right\}=\varnothing$