Chapter 2 Section 1 Question 6
Question
Let $S={\{1,2,3,4\}}$, $X=\textit{I}_{\{1,2\}}$, $Y=\textit{I}_{\{2,3\}}$, $Z=\textit{I}_{\{3,4\}}$. Let $W=X+Y+Z$
  1. Compute $W(1)$
  2. Compute $W(2)$
  3. Compute $W(4)$
  4. Determine whether or not $W \geq Z$
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 14 2020
(a) Compute $W(1)$
$$W(1)=X(1)+Y(1)+Z(1)$$ $$W(1)=1+0+0$$ $$W(1)=1$$
(b) Compute $W(2)$
$$W(2)=X(2)+Y(2)+Z(2)$$ $$W(2)=1+1+0$$ $$W(2)=2$$
(c) Compute $W(4)$
$$W(4)=X(4)+Y(4)+Z(4)$$ $$W(4)=0+0+1$$ $$W(4)=1$$
(d) Determine whether or not $W\geq Z$
We will start by computing W for $W(3)$ so we can know its value for our entire sample space $$W(3)=X(3)+Y(3)+Z(3)$$ $$W(3)=0+1+1$$ $$W(3)=2$$ Now we will compare W to Z for every element in our sample space
$W(1)\geq Z(1)\rightarrow 1 \geq 0$ True
$W(2)\geq Z(2) \rightarrow 2 \geq 0$ True
$W(3)\geq Z(3) \rightarrow 2 \geq 1$ True
$W(4)\geq Z(4) \rightarrow 1 \geq 1$ True
since our relation holds true for all elements of our sample space we can conclude that the relation is True
alternatively we could have concluded that the range of W is $[1,3]$ and the range of Z is [0,1] therefore $W\geq Z$.