“President Trump became the first president to talk about the importance of religious freedom at the United Nations, giving hope to people of faith around the world.”
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— Melania Trump, the first lady

The number of high-ranking women in Mr. Trump’s administration is not “unprecendented.” Melania Trump’s remark goes a step further than a comment made by a narrator in a video earlier in the program.

Of the 877 key executive branch nominees put forth by Mr. Trump for positions that require Senate confirmation, about 27.6 percent have been women. Of the 679 that have been confirmed, about 25 percent have been women.

That’s a lower percentage than the appointments of both former President Bill Clinton (37 percent) and former President Barack Obama (43 percent).

At the cabinet level, Mr. Trump has nominated seven women and 32 men. That’s lower than the eight women who served at cabinet-level positions in Mr. Obama’s first term and 10 in his second term, and the nine who served in Mr. Clinton’s second term.

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— Mike Pompeo, the secretary of state

President Trump has weakened NATO. His constant criticism of the alliance has generated widespread distrust of Mr. Trump among its European members and questions over the reliability of the United States. European allies are incensed over Mr. Trump’s denunciations of them, and they are aware the American president wants to withdraw from NATO, as several of Mr. Trump’s former senior aides have said.

In this statement, Mr. Pompeo is also trying to portray Mr. Trump as being tough on Russia. The opposite is true. A congressional inquiry last year found that Mr. Trump tried to hold up American weapons shipments to Ukraine, which is fighting against Russia-backed insurgents, in exchange for personal political favors. That resulted in the Democratic-led House of Representatives bringing two impeachment charges against Mr. Trump, though a Republican-led Senate found him not guilty on both counts. Mr. Trump has also praised President Vladimir V. Putin of Russia and said he believes Mr. Putin over his own intelligence agencies on Russia’s interference in the 2016 U.S. elections. And Mr. Trump has said nothing to Mr. Putin about U.S. intelligence findings that Russia offered bounties to Afghan fighters for the killing of American troops, which became public after The New York Times reported on the findings this summer.

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— Melania Trump, first lady

Other presidents have addressed the United Nations on the topic of religious freedom, including President Barack Obama in this address.

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— Secretary of State Mike Pompeo

President Trump did begin high-level diplomatic talks with North Korea. Kim Jong-un, the leader of North Korea, has not conducted nuclear tests or long-range missile tests since the two first met in Singapore in June 2018.

But national security hawks on North Korea say Mr. Trump’s concessions have weakened Washington’s leverage over Pyongyang and have emboldened Mr. Kim to take aggressive actions that have increased tensions in Asia. They criticize Mr. Trump for his meetings with Mr. Kim and for certain actions of his that have softened the American line on North Korea, including halting large-scale U.S. military exercises with South Korea after the Singapore summit to accede to the wishes of Mr. Kim.

Last year, Mr. Kim stepped up the pace of the testing of short-range missiles, which has worried Japan and South Korea. Mr. Kim has also said he no longer wants to pursue diplomacy with the United States and has denounced the U.S. government’s approach.

Mr. Trump has met with Mr. Kim three times — the only times that an American president has done so with a leader of North Korea since the Korean War — but those talks have done nothing to diminish North Korea’s nuclear weapons program. Experts say North Korea can produce enough fissile material for a nuclear warhead every six months.

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— Mike Pompeo, the secretary of state

Mr. Pompeo’s comments lack nuance. While it is true that officials in the city of Wuhan — where the coronavirus was first detected — tried to hide information from China’s central leadership, top Chinese officials do not appear to have engaged in deceit about the true threat posed by the virus to the extent portrayed by the Trump administration. The spread of the virus out of China was also a natural consequence of intercontinental travel; entry of the coronavirus into the United States was, ultimately, a repeat affair, with several large outbreaks seeded by individuals returning from Europe, not Asia. And the fault of the subsequent spread of the virus within U.S. borders certainly cannot be pinned on China. In just a few short months, the United States outstripped all other countries in coronavirus case counts — which currently exceed 5.7 million — and has clocked more than 178,000 deaths. China, with a population four times the size of the United States’, has yet to hit 100,000 cases or 5,000 deaths.

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— Mike Pompeo, the secretary of state

President Trump signed a limited trade agreement with China in January, which was the first such agreement ever signed between the two countries. It opened up China’s financial and agricultural markets and required China to make significant purchases of American products, but it was widely criticized for failing to address structural issues in the Chinese economy that have harmed American workers. As a result, most economists believe the trade pact will have a limited impact on American employment.

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— President Trump

Women’s unemployment rate did fall to its lowest levels since the 1950s in 2019 and early 2020, and last year, women accounted for about 1.5 million of the 2.1 million jobs gained — slightly more than 70 percent.

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— Eric Trump, President Trump’s son

Mr. Biden has called for increasing taxes on high-earning individuals and on corporations, and his tax proposals would increase federal revenues by $4 trillion over a decade, according to an analysis by the Tax Policy Center in March. About half of that total would come from increased taxes on businesses, and the other half would come from increased taxes on households with high incomes.

Other analyses have made similar projections for how much additional revenue Mr. Biden’s tax plans would generate, so the total figure that Mr. Trump cited is sound. But the tax increases Mr. Biden is proposing would not fall broadly across American households, as Mr. Trump suggested. Mr. Biden is not proposing to directly raise taxes on middle-class households. He reiterated in an interview that aired Sunday on ABC that he would not raise taxes on people who make below $400,000. That said, he wants to increase corporate taxes, and some portion of the burden of corporate taxes falls on workers — though it is debatable how much. The analysis by the Tax Policy Center, which took into account the indirect effects of raising corporate taxes, found that Mr. Biden’s proposals would reduce after-tax incomes across income groups. But the analysis found only a modest effect on most households, with the additional tax burden falling overwhelmingly on the highest earners. Nearly three-quarters of the proposed tax increase would fall on the top 1 percent.

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— Ryan Holets, a police officer in New Mexico

While overdose deaths did drop by around 5 percent in 2018, the first decrease since 1990, they started rising again last year. The decline was largely explained by reductions in deaths from prescription opioid painkillers, the drugs that started the nation’s addiction epidemic three decades ago. But deaths involving illicit drugs, particularly fentanyl, methamphetamine and cocaine, continued to increase, not just in 2018 but also last year. And preliminary data suggests 2020 could see even more overdose deaths, including in some states where overall drug deaths had started to decline.

President Trump and Congress have provided several billion dollars in grants to states since 2017 for treatment, prevention and recovery services, expanding access to medications that treat opioid addiction in particular. But at the same time, the administration has continued fighting in court to overturn the Affordable Care Act, a law that has allowed states to expand Medicaid and provide free addiction treatment to low-income adults.

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— Eric Trump, the president’s son

Former Vice President Joseph R. Biden Jr. has repeatedly said that he opposes defunding the police and instead has proposed “to get police more money.” He has proposed that federal aid to police departments could be contingent on whether agencies meet certain standards.

Mr. Biden has also said he supports the Second Amendment and is a gun owner himself. But Mr. Biden also argues for a ban on the sale of assault weapons and high-capacity magazines, and he supports background checks and other regulations.

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— Eric Trump

The president has made this claim dozens of times and it remains false. The $1.5 trillion tax cut, enacted in December 2017, ranks below at least half a dozen others by several metrics. The 1981 Reagan tax cut is the largest as a percentage of the economy and by its reduction to federal revenue. The 2012 Obama cut amounted to the largest cut in inflation-adjusted dollars: $321 billion a year.

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— Eric Trump, President Trump’s son

By most measurements, it is not true that the economy hit heights it had never seen before under President Trump. Before the pandemic, which sent the economy into a tailspin, overall growth was little changed under his administration — and much slower than the rates it had regularly achieved in the 20th century. The job market was strong, but that was a continuation of trends well underway during President Barack Obama’s administration. For instance, unemployment had continued a yearslong decline — one that started in 2009 — and stood at a half-century low of 3.5 percent in February 2020. It is true that Black American and Hispanic Americans were enjoying record low joblessness before the pandemic, but women actually experienced their lowest unemployment rates in the 1950s.

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— A video produced for the RNC

Kellyanne Conway, Mr. Trump’s third campaign manager for his 2016 bid, was the first woman to run a winning presidential campaign, and three of his White House press secretaries have been women. But the number of high-ranking women in his administration is not record-breaking.

Of the 877 key executive branch nominees put forth by Mr. Trump for positions that require Senate confirmation, about 27.6 percent have been women. Of the 679 that have been confirmed, about 25 percent have been women.

That’s a lower percentage than the appointments of both former President Bill Clinton (37 percent) and former President Barack Obama (43 percent).

At the cabinet level, Mr. Trump has nominated seven women and 32 men. That’s lower than the eight women who served at cabinet-level positions in Mr. Obama’s first term and 10 in his second term, and the nine who served in Mr. Clinton’s second term.

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— President Trump

The president’s friendly words about immigrants at the naturalization ceremony stands in stark contrast to almost four years in which he has repeatedly pursued anti-immigrant policies, often fueled by xenophobic language.

The president has largely blocked asylum seekers and refugees fleeing persecution, war and violence. He has built nearly 300 miles of border wall (though without getting Mexico to pay for it). He has made it harder for poor people to immigrate to the United States, imposed travel bans on mostly-Muslim countries, and separated migrant children from their parents at the border.

At times, his has used racist rhetoric, condeming “shithole countries” and complaining that people from Haiti “have AIDS.”

Even as he praised the new citizens on Tuesday, Mr. Trump has long sought to reduce legal immigration into the United States and has recently moved to shrink or eliminate visa programs that allow companies to hire foreigners to work in America. Aides to the president brag about the reductions in overall immigration, saying the efforts are helping to protect Americans from having to compete with immigrants for jobs.

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— Pam Bondi, former attorney general of Florida

Hunter Biden did in fact travel with his father aboard Air Force Two to China, and Hunter Biden did meet with a Chinese business partner during the trip. Several days after the trip, a Chinese government-linked private equity fund in which Hunter Biden has been involved, BHR Equity Investment Fund Management Co., won a business license from the Chinese government.

Hunter was on the board of the fund when it was formed in late 2013, and he later invested roughly $420,000, giving him a 10 percent stake, after his father had left the vice presidency. But Hunter’s lawyer has said that he has never been paid for his role on the board, and has not profited financially since he began as a part-owner. Hunter left the board in April, according to a letter produced by his lawyer. But as of June, he still owned his stake in the fund, which he was trying to sell. His lawyer did not respond to a request for comment about the status of that effort.

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— Pam Bondi, former attorney general of Florida

Mr. Biden did make those remarks on China in May 2019. They prompted criticism at the time from Republicans and some Democrats, including Senator Bernie Sanders of Vermont, who was Mr. Biden’s main challenger in the Democratic primary this year.

Mr. Biden’s aides say his comment was the kind of expression of confidence in the superiority of the United States that he has made throughout his political career. And Mr. Biden has made more recent comments on China that portray it as a formidable challenge. He told The Washington Post this year that China is the greatest “medium-term” strategic challenge for the United States. In a Foreign Affairs essay this year, he wrote: “China represents a special challenge. I have spent many hours with its leaders, and I understand what we are up against.”

He added: “The United States does need to get tough with China. If China has its way, it will keep robbing the United States and American companies of their technology and intellectual property.”

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— Abby Johnson, anti-abortion activist

Margaret Sanger, the sex educator who opened the first birth control clinic in the United States in 1916, did find common ground with those who advocated eugenics, the widely debunked theory that the human race could be bettered by encouraging people with traits like intelligence and hard work to reproduce. But there is no evidence that she was a racist who intended to “eradicate the minority population.”

Ms. Sanger’s views on race and eugenics have been widely debated. N.P.R. fact-checked similar claims made by Ben Carson in a 2015 interview with Fox News.

In the United States, eugenics intersected with the birth control movement in the 1920s, and Ms. Sanger reportedly spoke at eugenics conferences. She also talked about birth control being used to facilitate “the process of weeding out the unfit [and] of preventing the birth of defectives,” N.P.R. wrote, adding, “Historians seem to disagree on just how involved in the eugenics movement she was. Some contend her involvement was for political reasons — to win support for birth control.”

In 1939, Ms. Sanger founded what she called “the Negro Project,” one of the first major undertakings of the Birth Control Federation of America, a forerunner to Planned Parenthood. It was aimed at giving Black women control over their own fertility and limiting reproduction among poor and uneducated women who could least afford it. Conservatives have argued that it was a racist effort to reduce the Black population. The Margaret Sanger Papers Project, a New York University initiative to examine Sanger’s writings, addressed the project and dismissed such assertions in a 2001 newsletter.

“In fact, the Negro Project did not differ very much from the earlier birth control campaigns in the rural South designed to test simpler methods on poor, uneducated and mostly white agricultural communities,” the papers project wrote. “Following these other efforts in the South, it would have been more racist, in Sanger’s mind, to ignore African-Americans in the South than to fail at trying to raise the health and economic standards of their communities.”

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— Pam Bondi, former Florida attorney general

It is true that Vice President Joseph R. Biden Jr.’s son Hunter, despite having no discernible experience in Ukraine or the energy sector, was paid as much as $50,000 a month in some months to serve on the board of the directors of a Ukrainian gas company owned by an oligarch who was widely considered in the international community to be corrupt. And it is true that Vice President Biden, the Obama administration’s point-man on Ukraine, helped force the ouster of a Ukrainian prosecutor, partly by threatening to withhold $1 billion in United States loan guarantees if Ukraine’s leaders did not dismiss the prosecutor. But there is no evidence that the prosecutor was aggressively pursuing investigations into the oligarch or his gas company. Instead, the oligarch’s allies say that the prosecutor was using the threat of prosecution to try to solicit bribes from the oligarch and his team — allegations that comport with others that swirled around the prosecutor, who was eventually fired. The pressure campaign to have the prosecutor fired was international, embraced by many Western governments, not just the Obama administration.

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— Cissie Graham Lynch, a granddaughter of Billy Graham, the founder of Christianity Today

While there is plenty of room for subjective judgments about what constitutes religious freedom and there are deep divisions between left and right about how politics should intersect with faith, Mr. Biden is a practicing Catholic. There were opening prayers during the Democratic National Convention. And one of Mr. Biden’s close friends, Senator Chris Coons, offered a lengthy testimony to Mr. Biden’s faith, saying Mr. Biden would be “a president for Americans of all faiths, as well as people of conscience who practice no particular faith.”

Ms. Graham Lynch was likely referring to positions Mr. Biden has — such as his support for same-sex marriage or abortion rights — that she personally views as against her faith.

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— Abby Johnson, anti-abortion activist

It is true that Mr. Trump has made a huge number of appointments to the bench — 203, including two to the nation’s highest court. And it is true that his appointees have been confirmed with polarizing paper trails, having spent their careers more openly engaged in causes important to Republicans, such as fighting against government funding for abortion, than the appointees of past presidents. But judges do not take the bench as “pro-life” or “pro-choice,” and, upon questioning from senators in confirmation hearings, many of Mr. Trump’s appointees explicitly pledged to apply existing precedent like Roe v. Wade, the landmark ruling that legalized abortion.