EDITOR'S NOTE — A look at political claims that take shortcuts with the facts or don't tell the full story

WASHINGTON (AP) — A look at some of the claims in the latest Republican presidential debate and how they compare with the facts:

JEB BUSH: "We need to embed our troops in the Iraqi military."

THE FACTS: The U.S. is already doing that.

U.S. special forces are working side by side with Iraqi forces in the fight against Islamic State militants and American military advisers and trainers are working with Iraqi troops in various locations. To be sure, Bush has called for an intensification of the military effort in a variety of ways, but debate viewers would not know from his comment that U.S. troops are already operating with Iraqi and Kurdish forces.

His comment fits a pattern in the Republican race as a number of candidates criticize President Barack Obama's course against ISIS while proposing largely the same steps that are already underway.

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TED CRUZ: "You would carpet bomb where ISIS is, not a city."

THE FACTS: The Texas senator's conviction that the Islamic State group can be routed with an air campaign of overwhelming force is hard to square with the reality on the ground. ISIS fighters are holed up in a variety of cities, amid civilians, raising questions about how he could direct a carpet bombing that only singles out the enemy.

He was asked in the debate if he'd be willing to cause civilian casualties in Raqqa, a major Syrian city that has become de facto capital of the Islamic State group's so-called caliphate. ISIS is also in control of the Iraqi city of Mosul.

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DONALD TRUMP: "Our country is out of control. People are pouring through the southern border."

THE FACTS: Arrest statistics are widely regarded as the best measure, if an imperfect one, of the flow of people crossing illegally into the U.S. And Trump's suggestion that illegal immigration is increasing at the border is not supported by arrest statistics discussed in recent months by Homeland Security Secretary Jeh Johnson.

Johnson has said that during the 2015 budget year that ended in September, about 330,000 people were caught crossing the Mexican border illegally, a near 40-year low in border arrests. During the 2014 budget year, roughly 486,000 people were arrested.

In recent months there has been a spike in the arrests at the border, but primarily of children traveling alone and families, mostly from Central America.

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CRUZ: "And even worse, President Obama and Hillary Clinton are proposing bringing tens of thousands of Syrian refugees to this country when the head of the FBI has told Congress they cannot vet those refugees."

THE FACTS: Cruz repeated inflated estimates of how many Syrian refugees the Obama administration plans to admit to the United States. Obama has announced plans to resettle about 10,000 refugees in the next year.

The vetting process for refugees takes, on average, about two years and is routinely longer for refugees from Syria and Iraq. The administration has said refugees being considered for resettlement in the United States are subject to additional scrutiny. The administration has declined to describe what the scrutiny involves, saying it is classified.

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CARLY FIORINA, speaking of security threats to the U.S.: "We need the private sector's help because the government is not innovating, technology is running ahead by leaps and bounds...They must be engaged and they must be asked. I will ask them."

THE FACTS: They've been asked.

The Obama administration has been in discussions with technology companies, especially in Silicon Valley, over the last year about the use of encrypted communications and how the government can penetrate them for national security purposes. After the attack in San Bernardino, California, Obama again said he would urge high-tech and law enforcement leaders to make it harder for terrorists to use technology to escape justice.

That's not to say the effort has been effective. But as in the case of candidates talking about the campaign against ISIS, Fiorina pitches something that is in motion.

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CRUZ: "We didn't monitor the Facebook page of the San Bernardino terrorist because DHS thought it would be inappropriate."

THE FACTS: The Department of Homeland Security is able to look at social media case by case, but it would be unwieldy to do such searches on everyone applying for a visa and it's unclear whether looking at the Facebook pages of the shooters in the California attacks would have prevented them.

While attacker Tashfeen Malik, 29, did use social media and spoke online of martyrdom and jihad, Rep. Adam Schiff, D-Calif., has said such posts weren't public. Facebook did find a profile under an alias linked to Malik with a post pledging the allegiance of her husband, 28-year-old Syed Farook, and herself to the leader of the Islamic State group minutes before the attack. Facebook notified law enforcement.

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Associated Press writer Alicia A. Caldwell contributed to this report.