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Understanding Statements, Negations, and Quantified Statements in Logic

This section delves into the fundamentals of logical statements, negations, and quantified statements. A statement is defined as a sentence that can be true or false, but not both. Negation refers to the creation of a statement that opposes the truth value of the original statement. Moreover, quantified statements express attributes about all, some, or none of a particular set. Exercises challenge readers to apply these concepts through practical examples, emphasizing how to negate statements and use symbolic notation effectively.

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Understanding Statements, Negations, and Quantified Statements in Logic

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  1. Thinking Mathematically Logic 3.1 Statements, Negations, and Quantified Statements

  2. Statements • A “statement” is a sentence that is either “true” or “false” but not both at the same time. Exercise Set 3.1 #3, 9 Statement? • On January 20, 2009, John McCain became America’s 44th president. • Is the unexamined life worth living? • 9 + 6 = 16.

  3. Statements - “Negation” • The “negation” of a statement is another statement that has the opposite “truth value.” That is when a statement is true its negation is false and when the statement is false its negation is true. Exercise Set 3.1 #15 Form the negation of “It is raining” What is the negation of 2+2 = 5

  4. Statements - “Symbolism” Just as x can be used as a name for a number, a symbol such as p can be used as a name for a statement. • When p is used as a name for a statement the symbols ~p are used as a name for the negation of p.

  5. Exercise Set 3.1 #23 p: One works hard q: One succeeds r: The temperature outside is not freezing s: It is not true that the heater is working The temperature outside is freezing = ? Exercise Set 3.1 #27 p: Listening to classical music makes infants smarter. q: Subliminal advertising makes you buy things. r: Sigmund Freund’s father was not 20 years older than his mother. s: Humans and bananas do not share approximately 60% of the same DNA structure. ~r = ? Examples: Using Symbols for Statements

  6. “Quantified” Statements • A “quantified” statement is one that says something about “all”, “some”, or “none” of the objects in a collection. Exercise Set 3.1 #29, #31, #35 For each of the statements: • All whales are mammals. • Some students are business majors. • No Democratic presidents have been impeached. • Give an equivalent statement • Negate the statement.

  7. Thinking Mathematically Logic 3.1 Statements, Negations, and Quantified Statements

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