To Do Wudu Or Not to Do Wudu?

Is wudu required after urinating?

2.2 Things which break wudu'

2.2a. Excreta which oblige wudu'

You have to do wudu' after urinating or defecating or passing wind.

[This clarifies what excreta which make wudu' necessary: urine from the front orifice and faeces from the rear orifice and passing wind, which designates wind which emerges from the anus, whether or not with a sound. As for wind which emerges from the penis or vagina, it does not require wudu' since it is not considered as one of things which break wudu'.] RisWudu2 (Page 1)

Regarding farting, see my post Farting

It was narrated that Abu Hayyah al-Wadi`i said:

I saw ‘Ali (رضي الله عنه) urinate in ar-Rahbah, then he called for water and did wudoo’. He washed his hands three times, rinsed his mouth and nose three times, washed his face three times, washed his forearms three times each, wiped his head and washed his feet three times each. Then he said: I saw the Messenger of Allah (ﷺ) do as you have seen me do.

https://sunnah.com/ahmad:1350

Ali did wudu after relieving himself.

Yahya related to me from Malik from Nafi that Abdullah ibn Umar urinated in the market place and then did wudu, washing his face and hands and wiping his head. Then as soon as he had come into the mosque, he was called to pray over a dead person, so he wiped over his leather socks and prayed.

Muwatta Malik Book 2, Hadith 44

Ibn Umar did wudu after urinating in the market place.

Narrated Ibn 'Abbas:

"The Messenger of Allah (ﷺ) cae out from the toilet and some food was brought to him. They said: 'Shall we bring you some water for Wudu'?' He said: 'I have only been ordered to perform Wudu' when standing for Salat.'"

[Abu 'Eisa said:] This Hadith is Hasan [Sahih]. 'Amr bin Dinar has reported it from Sa'eed bin Al-Huwairith, from Ibn 'Abbas. 'Ali bin Al-Madini said: 'Sufyan Ath-Thawri disliked washing the hands before eating food, and he disliked placing the bread under the bowl.'"

https://sunnah.com/tirmidhi:1847

It was narrated from Abu Hurairah that the Messenger of Allah (ﷺ) went out to toilet, then food was brought. A man said:

“O Messenger of Allah, are you not going to perform ablution?” He said: ‘Am I going to pray?’

Sunan Ibn Majah 3261

Allah's boss didn't do wudu after urinating or defecating.