Closed Under Addition

A set is closed under addition if the sum of two elements in a set is also a member of the same set. $\vec{x},\vec{y}\in S⟹\vec{x}+\vec{y}\in S$

Examples

The set $P=\{(x,y,z)\in R^3;z=0\}$ Is closed under addition. We can verify this by:

• $(x_1,y_1,0)$ and $(x_2,y_2,0)$ have a sum $(x_1+x_2,y_1+y_2,0)$ which is in $P$ The set $P=\{(x,y,z)\in R^3;z=1\}$ Is not closed under addition.
• $(1,1,1)$ and $2,1,1$ are in $P$, but $(1,1,1)+(2,1,1)=(3,3,2)$ which is not in $P$.

Proving Closed Under Addition Example