Answer
$yes\,\,,\left \{ E,O \right \}\,is\,a\,partition\,\,of\,\,\mathbb{Z}.$
$E\,\,be\,\,the\,\,set\,\,of\,\,all\,\,even\,\,integers\,\,,\,\,O\,\,the\,\,set\,\,of\,\,all\,\,odd\,\,
integers\,\,and\,\,\mathbb{Z}\,the\,set\,of\,\,all\,\,integers$
Work Step by Step
$E\,\,be\,\,the\,\,set\,\,of\,\,all\,\,even\,\,integers\,\,,\,\,O\,\,the\,\,set\,\,of\,\,all\,\,odd\,\,
integers\,\,and\,\,\mathbb{Z}\,the\,set\,of\,\,all\,\,integers$
$E=\left \{ 0,2,4,6,.... \right \}\,\,,O=\left \{ 1,3,5,7,.... \right \}
$
$E\cup O=\mathbb{Z} (as\,\,every\,\,integer\,\,either\,\,odd\,\,or\,\,even\,)$
$\,\,and\,\,E\cap O=\varnothing \,\,(as\,any\,integer\,\,can\,\,not\,\,be\,both\,odd\,and\,even)$
$so \left \{ E,O \right \}\,is\,a\,partition\,\,of\,\,\mathbb{Z}$