Question

⋃ 𝐴𝑖

∝

1 and ⋂ 𝐴𝑖

∞

1

a) 𝐴𝑖 = {𝑖, 𝑖 + 1,𝑖 + 2, … }

b) 𝐴𝑖 = {0,𝑖}

c) 𝐴𝑖 = {−𝑖, − 𝑖 + 1, … , −1, 0, 1, … , 𝑖 − 1, 𝑖}

d) 𝐴𝑖 = {−𝑖, 𝑖}

Answer 1

Let us find $\bigcup\limits_{i=1}^{\infty} A_i$ and $\bigcap\limits_{i=1}^{\infty} A_i$ for the following sets $A_i.$

a) If $𝐴_𝑖 = \{𝑖, 𝑖 + 1,𝑖 + 2, … \}$ then $\bigcup\limits_{i=1}^{\infty} A_i=A_1=\mathbb N$ and $\bigcap\limits_{i=1}^{\infty} A_i=\emptyset.$

b) For $𝐴_𝑖 = \{0,𝑖\}$ we get that $\bigcup\limits_{i=1}^{\infty} A_i=\{0,1,2,...\}=\mathbb N_0$ and $\bigcap\limits_{i=1}^{\infty} A_i=\{0\}.$

c) If $𝐴_𝑖 = \{−𝑖, − 𝑖 + 1, … , −1, 0, 1, … , 𝑖 − 1, 𝑖\}$ then $\bigcup\limits_{i=1}^{\infty} A_i=\mathbb Z$ and $\bigcap\limits_{i=1}^{\infty} A_i=A_1=\{-1,0,1\}.$

d) For $𝐴_𝑖 = \{−𝑖, 𝑖\}$ we get that $\bigcup\limits_{i=1}^{\infty} A_i=\mathbb Z\setminus\{0\}$ and $\bigcap\limits_{i=1}^{\infty} A_i=\emptyset.$