The document summarizes the closure properties of addition and multiplication for several sets. It finds that: (1) The set {0} is closed under both addition and multiplication as 0 + 0 = 0 x 0 = 0, which is in the set. (2) The set {0, -1} is not closed under addition or multiplication as combinations such as -1 + -1 = -2 are not in the set. (3) The document prompts the reader to check closure for the sets {1} and {1, -1} themselves.

1. Solution of Ex. 1.1

2. Q # 1
Does the following sets have closure property w.r.t. addition and multiplication
(i)
{ 0 }
sol.
Let A = { 0 }
0 + 0 = 0 ∈ A
So A = { 0 } is close w.r.t. addition
Now
0 x 0 = 0 ∈ A
so A= { 0 } is close w.r.t. multiplication

3. ( ii ) {1 } Do your self
( iii )
𝐴 = 0 , −1
GRW 2013 , BWP 2014 , MTN , BWP , LHR 2017 , DGK ,
RWP 2019 , RWP 2019
+ 0 −𝟏
0 0 -1
-1 -1 -2
-2 ∉ 𝐴, so this set is not close w.r.t.
Addition
x 0 −𝟏
0 0 0
-1 0 1
1 ∉ 𝐴 , so set A is not close w.r.t. X

4. (iv) { 1 , -1 } Do your self
MTN 2013 , LHR , RWP , BWP 2014 , SWL ,
SGD 2017 , MTN 2018 , GRW , SGD 2019