# Two dice are thrown.The events A,B and C are as follows :A : getting an even number in first die,B : getting an odd number on the first die,C : getting the sum of the numbers on the dice $\leq$ 5.State true or false : A and B are mutually exclusive & exhaustive

$\begin{array}{1 1}(A)\;\text{True}\\(B)\;\text{False}\end{array}$

Given two dice are thrown
Possible outcomes =$6\times 6=36$
$\therefore$ The sample space
$S=\left\{\begin{array}{1 1}(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)\\(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)\\(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)\\(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)\\(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)\\(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)\end{array}\right\}$
A: Getting an even number on first die
B: Getting an odd number on first die
A=$\left\{\begin{array}{1 1}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$
B=$\left\{\begin{array}{1 1}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\end{array}\right\}$
$A\cap B=\phi$
$A\cup B=S$
$A$ and $B$ are grouped together forms the sample space.
$\therefore$ A and B are mutually exclusive & exhaustive
Hence the given statement is True.