Two dice are thrown. Let $E$be the event that the sum of the dice is odd, let $F$be the event that at least one of the dice lands on $1$, and let $G$be the event that the sum is $5$. Describe the eventslocalid="1649252717741" $EF,E\cup F,FG,E{F}^c,andEFG$.

Let $x_1$be the number that appears on first dice and $x_2$be the number that appears on second dice .

$E=\left\{\left(x_1,x_2\right):x_1+x_2=odd\right\}$

$F=\left\{\left(1,x_2\right),\left(x_1,1\right)\right\}$

role="math" localid="1649251598431" $$\begin{gathered} E=\left\{\left(1,2\right),(1,4),(1,6),(2,1),(2,3),(2,5),\left(3,2\right),(3,4),(3,6), \\ (4,1),(4,3),(4,5),\left(5,2\right),(5,4),(5,6),(6,1),(6,3),(6,5)\right\} \\ F=\left\{\left(1,1\right),(1,2),(1,3),(1,4),(1,5),(1,6),\left(2,1\right), \\ (3,1),(4,1),(5,1),(6,1)\right\} \end{gathered}$$

Therefore,$EF=\left\{(1,2),(1,4),(1,6),(2,1),(4,1),(6,1)\right\}$

Let $x_1$be the number that appears on first dice and $x_2$be the number that appears on second dice

role="math" localid="1649252815561" $E=\left\{\left(x_1,x_2\right):x_1+x_2=odd\right\}$

$F=\left\{\left(1,x_2\right),\left(x_1,1\right)\right\}$

$$\begin{gathered} E=\left\{\left(1,2\right),(1,4),(1,6),(2,1),(2,3),(2,5),\left(3,2\right),(3,4),(3,6), \\ (4,1),(4,3),(4,5),\left(5,2\right),(5,4),(5,6),(6,1),(6,3),(6,5)\right\} \\ F=\left\{\left(1,1\right),(1,2),(1,3),(1,4),(1,5),(1,6),\left(2,1\right), \\ (3,1),(4,1),(5,1),(6,1)\right\} \end{gathered}$$

Therefore,role="math" localid="1649252729720" $$\begin{gathered} E\cup F=\left\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,5),(3,1) \\ ,(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),(5,1),(5,2), \\ (5,4),(5,6),(6,1),(6,3),(6,5)\right\} \end{gathered}$$

Let $x_1$be the number that appears on first dice and $x_2$be the number that appears on second dice

$F=\left\{\left(1,x_2\right),\left(x_1,1\right)\right\}$

$$\begin{gathered} F=\left\{\left(1,1\right),(1,2),(1,3),(1,4),(1,5),(1,6),\left(2,1\right), \\ (3,1),(4,1),(5,1),(6,1)\right\} \end{gathered}$$

localid="1649253609583" $G=\left\{\left(x_1,x_2\right):x_1+x_2=5\right\}$

localid="1649253631472" $G=\left\{\left(1,4\right),(2,3),(3,2),(4,1)\right\}$

Therefore,$FG=\left\{(1,4),(4,1)\right\}$

Let $x_1$be the number that appears on first dice and $x_2$be the number that appears on second dice.

$E=\left\{\left(x_1,x_2\right):x_1+x_2=odd\right\}$

$F=\left\{\left(1,x_2\right),\left(x_1,1\right)\right\}$

role="math" localid="1649253358260" $$\begin{gathered} E=\left\{\left(1,2\right),(1,4),(1,6),(2,1),(2,3),(2,5),\left(3,2\right),(3,4),(3,6), \\ (4,1),(4,3),(4,5),\left(5,2\right),(5,4),(5,6),(6,1),(6,3),(6,5)\right\} \\ F=\left\{\left(1,1\right),(1,2),(1,3),(1,4),(1,5),(1,6),\left(2,1\right), \\ (3,1),(4,1),(5,1),(6,1)\right\} \end{gathered}$$

role="math" localid="1649253518201" $$\begin{gathered} F^c=S-F \\ =\left\{\left(2,2\right),(2,3),(2,4),(2,5),(2,6),\left(3,2\right),(3,3),(3,4),(3,5),(3,6), \\ \left(4,2\right),(4,3),(4,4),(4,5),(4,6),\left(5,2\right),(5,3),(5,4),(5,5),(5,6), \\ \left(6,2\right),(6,3),(6,4),(6,5),(6,6),\right\} \end{gathered}$$

Therefore,$E{F}^c=\left\{(2,3),(2,5),(3,2),(3,4),(3,6),(4,3),(4,5),(5,2),(5,4),(5,6),(6,3),(6,5)\right\}$

Let $x_1$be the number that appears on first dice and

$x_2$be the number that appears on second dice

$E=\left\{\left(x_1,x_2\right):x_1+x_2=odd\right\}$

$F=\left\{\left(1,x_2\right),\left(x_1,1\right)\right\}$

$G=\left\{\left(x_1,x_2\right):x_1+x_2=5\right\}$

localid="1649253644236" $$\begin{gathered} E=\left\{\left(1,2\right),(1,4),(1,6),(2,1),(2,3),(2,5),\left(3,2\right),(3,4),(3,6), \\ (4,1),(4,3),(4,5),\left(5,2\right),(5,4),(5,6),(6,1),(6,3),(6,5)\right\} \\ F=\left\{\left(1,1\right),(1,2),(1,3),(1,4),(1,5),(1,6),\left(2,1\right), \\ (3,1),(4,1),(5,1),(6,1)\right\} \\ G=\left\{\left(1,4\right),(2,3),(3,2),(4,1)\right\} \end{gathered}$$

Therefore,$EFG=\left\{(1,4),(4,1)\right\}$