Chapter 2 Section 1 Question 7
Question
Let $S={\{1,2,3\}}$, $X=\textit{I}_{\{1\}}$, $Y=\textit{I}_{\{2,3\}}$, $Z=\textit{I}_{\{1,2\}}$. Let $W=X-Y+Z$
  1. Compute $W(1)$
  2. Compute $W(2)$
  3. Compute $W(3)$
  4. Determine whether or not $W \geq Z$
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 14 2020
(a) Compute $W(1)$
$$W(1)=X(1)-Y(1)+Z(1)$$ $$W(1)=1-0+1$$ $$W(1)=2$$
(b) Compute $W(2)$
$$W(2)=X(2)-Y(2)+Z(2)$$ $$W(2)=0-1+1$$ $$W(2)=2$$
(c) Compute $W(3)$
$$W(3)=X(3)-Y(3)+Z(3)$$ $$W(3)=0-1+0$$ $$W(3)=-1$$
(d) Determine whether or not $W\geq Z$
We will compare W to Z for every element in our sample space
$W(1)\geq Z(1)\rightarrow 2 \geq 1$ True
$W(2)\geq Z(2) \rightarrow 2 \geq 1$ True
$W(3)\geq Z(3) \rightarrow -1 \geq 0$ FALSE
since we have shown that the relation does not hold for atleast one element of our sample space, we can conclude that the relation is False
alternatively we could have concluded that the range of W is $[-1,2]$ and the range of Z is [0,1] therefore $W\ngeq Z$.