### 3.2 Statements & Negations

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• Math Help

The word "some" means "at least one." In Example 3, there were 4 presidential candidates who won the popular vote, but still lost the election. However, even if there were only one such person, the following statement would be true.

In the United States, some presidential candidates have won the popular vote, but still lost the election.

The set diagram shown in Example 3 is a classic use of a formal Venn diagram. There are two basic sets.

Set A: Candidates who won the popular vote

Set B: Candidates who won the election

These two basic sets determine four regions in the diagram. Moreover, each of the four regions has historical members.

• Consumer Suggestion

Did you know that Thomas Jefferson, the principal author of the Declaration of Independence, died on Independence Day? Click here to learn more interesting facts about the Presidents of the United States.

• Checkpoint Solution

"In the United States, 14 presidents served as vice presidents: J. Adams, Jefferson, Van Buren, Tyler, Fillmore, A. Johnson, Arthur, T. Roosevelt, Coolidge, Truman, Nixon, L. Johnson, Ford, and George H.W. Bush. Of these, Fillmore, A. Johnson, Arthur, and Ford did not win a presidential election."

Here is one way to draw a set diagram showing this information.

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