Question

Explain whether the following given sets are closed under addition:

a. B ={0}
b. T = {0, 3, 6, 9, 12, . . . }
c .N = {1, 2, 3, 4, 5, . . .}
d. V = {3, 5, 7}

Answer 1

Explanation:

a) Yes.

`\forall\ a,b\ \in\ B,\ a+b\in B\ (since\ 0+0=0)`

b) Yes

`W=\{0,1,2,3,..,\}\ (whole\ numbers)\\\forall\ a,b\ \in\ 3W,\ a+b\in 3W\\since:\\a\ \in\ 3W \Longrightarrow\ \exists\ k_1\in W | a=3*k_1\\\\b\ \in\ 3W \Longrightarrow\ \exists\ k_2\in W | b=3*k_2\\\\a+b=3*k_1+3*k_2=3*(k_1+k_2) \in 3W\\`

c) Yes

`\forall n\ \in\ \mathbb{N}: \Longrightarrow\ n+1 \in\ \mathbb{N}\\`

d) No

`3\in V , 3+3=6\\otin V\\`