JAMIA 2023 — Mathematics PYQ

The union collects every element from all sets $A_1,A_2,A_3,\dots$

• $A_1=\{-1,1\}$

• $A_2=\{-2,2\}$

• $A_3=\{-3,3\}$

  When you combine all these, you get all positive and negative integers. Since $0$ is not included in any $A_i$, the result is the set of all integers $\mathbb{Z}$ excluding zero.

  $$\bigcup _{i=1}^{\infty }{A}_i=\mathbb{Z}-\{0\}$$

2. The Intersection ($\cap$)

The intersection looks for elements that appear in every single set.

• Does $1$ appear in all sets? No, only in $A_1$.

• Does $2$ appear in all sets? No, only in $A_2$.

  Since there is no common element shared between $A_1,A_2,A_3,\dots$, the intersection is the empty set.

  $$\bigcap _{i=1}^{\infty }{A}_i=\varphi$$

Final Answer:

The union is $\mathbb{Z}-\{0\}$ and the intersection is $\varphi$.