Question

Explain whether the following given sets are closed under addition:

Answer 1

Answer:

Step-by-step 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

$3\in V , 3+3=6\notin V$