The county also bid farewell to outgoing Councilmembers Sally Elliott and John Hanrahan, who both decided not to run for reelection.
McMullin told those in attendance that one of her goals as a council member has always involved regional planning and cooperation.
"And I would like to see us continue to enrich and grow our relationships with our fellow citizens and our fellow public servants in the cities and counties around us," she said.
McMullin added that she would like to put the "acrimony" of the last year behind them, and move forward in a positive and cooperative manner.
Carson said she was excited to be there and to have so much support over the last several months, but added that, moving forward, the government needs to work together with the residents.
"I hope we can work together for the common good of Summit County," she said. "It is a government of the people and for the people, and I look forward to meeting people from all corners of the county."
Carson acknowledged that the County Council will face some difficult issues, such as the budget and county planning.
"So we've got to come together to work on them," she said.
Armstrong said it is humbling to sit up on the dais and be responsible for 39,000 people.
"Their hopes, their dreams, and the way they live in this county," he said. "And I hope to discharge that with honor and to live up to the trust of the people who put me here."
Armstrong also acknowledged the "significant" issues facing the county.
"I think growth is going to continue to be a challenge. We need to manage that, and we need to also find a way to bring the two halves of the county together and to restore trust to those who lack trust in the county government," he said.
County Council Chair Dave Ure expressed his appreciation for having served on the County Council with Elliott, acknowledging that although they didn't always see eye-to-eye, Elliott taught him that his viewpoint is not necessarily always right.
"And she's very ambitious and very energetic, and has worked very hard on all the projects she has undertaken in her life," Ure said. "And I don't imagine she'll quit now that she's going out of office. There will be goals and ideas she'll want to put across in the community and also in her family."
Elliott quipped that she'll miss serving on the council "very much, but not very often."
She added that she's not leaving the county and has already promised to volunteer time with Summit County Historical Society and the Historic Preservation Committee.
Outgoing County councilmembers
"So while I ski my little brains out, I'll still be available to help you guys with anything you need," she said.
Hanrahan, who let out a loud "whoop" as he approached the microphone, said that unlike Elliott, he would be unavailable to the government for a while. When asked to serve on a Park City commission two weeks ago, Hanrahan told them he "had no interest whatsoever."
"But I've enjoyed this. It's been great. And I'm ready to move on. What I really enjoyed most was working with this group of people, not just the council, but also the staff and various department heads. It was truly a phenomenal education for me, and I'll never regret it," he said.
Ure said Hanrahan has worked as hard as anyone to make sure there was a balance in the government, and between the east and west sides of the county.
"Over the last four years, I've learned a lot from John," he said. "And I greatly admire his ability to think, and his ability to stand up and state his opinion. It didn't matter if he took on the whole world, John was not afraid to voice his opinion on what he thought was right. And his principles were very clean and true. I don't know if John ever made a vote on a political issue. He made it on a principle issue."