{"id":5627,"date":"2021-03-29T02:54:17","date_gmt":"2021-03-29T02:54:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nwfsc-mathforliberalartscorequisite\/?post_type=chapter&p=5627"},"modified":"2021-07-07T02:07:21","modified_gmt":"2021-07-07T02:07:21","slug":"negating-statements","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-mathforliberalartscorequisite\/chapter\/negating-statements\/","title":{"raw":"Negating Statements","rendered":"Negating Statements"},"content":{"raw":"In the last section, we learned how to negate a conjunction and a disjunction using DeMorgan's Laws. Here, we will also learn how to negate the conditional and quantified statements.\r\n

Negating an Implication<\/h3>\r\nRecall that we learned about implications.\u00a0\u00a0Implications are logical conditional sentences stating that a statement\u00a0p<\/em>, called the antecedent, implies a consequence\u00a0q<\/em>.\u00a0 Also recall that the truth table for an implication has three trues and one false in the final column.\u00a0 The only place where the implication was false was when the antecedent is true and the consequent is false.\u00a0 So to negate an implication, we want that case to be true and all others to be false.\u00a0 So the negation of an implication is p\u00a0\u2227 ~q.\r\n
\r\n

Key Takeaways<\/h3>\r\nThe negation of p\u2192q is p\u00a0\u2227~q.\r\n\r\nOr in symbols, ~ (p\u2192q) =\u00a0p\u00a0\u2227~q.\r\n\r\n<\/div>\r\n

Quantified Statements<\/h3>\r\nWe use quantified statements often in mathematics, but they are also used in everyday life.\u00a0 An example of a quantified statement in math might be \"All even numbers are divisible by 2.\"\u00a0 An example of a non-mathematical quantified statement might be \"On every assignment, there is someone who forgets to complete the assignment.\"\r\n\r\nIn math, we use two symbols for quantified statements: the universal quantifier and the existential quantifier.\u00a0 The universal quantifier means \"for all\" or \"for every\".\u00a0 It is denoted by\u00a0\u2200.\u00a0 The existential quantifier means \"there exists\" or \"for some\".\u00a0 It is denoted by\u00a0\u2203.\r\n
\r\n

Examples<\/h3>\r\nDetermine if each statement uses the universal or existential quantifier.\r\n
    \r\n \t
  1. All mathematicians wear glasses.<\/li>\r\n \t
  2. There exists some number such that x2<\/sup> = x.<\/li>\r\n \t
  3. All politicians are dishonest.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"424820\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"424820\"]1. Universal 2. Existential 3. Universal[\/hidden-answer]\r\n\r\n<\/div>\r\n

    Negating Quantified Statements<\/h3>\r\nRecall that negating a statement changes its truth value.\u00a0 So negating a true statement gives a false statement and negating a false statement gives a true statement.\r\n
    \r\n

    Key Takeaways<\/h3>\r\nThe negation of a universal statement (All A are B) is an existential statement (There exists some A that is not B).\r\n\r\nThe negation of an existential statement (There exists some A that is B) is a universal statement (All A are not B).\r\n\r\n<\/div>\r\n
    \r\n

    Examples<\/h3>\r\nNegate each statement.\r\n
      \r\n \t
    1. All mathematicians wear glasses.<\/li>\r\n \t
    2. There exists some number such that x2<\/sup> = x.<\/li>\r\n \t
    3. All politicians are dishonest.<\/li>\r\n<\/ol>\r\n1: There exists at least one mathematician that does not wear glasses.\r\n\r\nThis is a universal statement.\u00a0 So the negation will be an existential statement.\u00a0 Notice that the original statement was false, and the negation is true.\r\n\r\n2: For all numbers, x2\u00a0<\/sup>\u2260 x.\r\n\r\nThis is an existential statement, so the negation will be a universal statement.\u00a0 Notice that the original statement is true (0 and 1 fit the property), and the negation is false.\r\n\r\n3: There exists at least one politician that is honest.\r\n\r\nThis is a universal statement.\u00a0 So the negation will be an existential statement.\r\n\r\n<\/div>\r\n ","rendered":"

      In the last section, we learned how to negate a conjunction and a disjunction using DeMorgan’s Laws. Here, we will also learn how to negate the conditional and quantified statements.<\/p>\n

      Negating an Implication<\/h3>\n

      Recall that we learned about implications.\u00a0\u00a0Implications are logical conditional sentences stating that a statement\u00a0p<\/em>, called the antecedent, implies a consequence\u00a0q<\/em>.\u00a0 Also recall that the truth table for an implication has three trues and one false in the final column.\u00a0 The only place where the implication was false was when the antecedent is true and the consequent is false.\u00a0 So to negate an implication, we want that case to be true and all others to be false.\u00a0 So the negation of an implication is p\u00a0\u2227 ~q.<\/p>\n

      \n

      Key Takeaways<\/h3>\n

      The negation of p\u2192q is p\u00a0\u2227~q.<\/p>\n

      Or in symbols, ~ (p\u2192q) =\u00a0p\u00a0\u2227~q.<\/p>\n<\/div>\n

      Quantified Statements<\/h3>\n

      We use quantified statements often in mathematics, but they are also used in everyday life.\u00a0 An example of a quantified statement in math might be “All even numbers are divisible by 2.”\u00a0 An example of a non-mathematical quantified statement might be “On every assignment, there is someone who forgets to complete the assignment.”<\/p>\n

      In math, we use two symbols for quantified statements: the universal quantifier and the existential quantifier.\u00a0 The universal quantifier means “for all” or “for every”.\u00a0 It is denoted by\u00a0\u2200.\u00a0 The existential quantifier means “there exists” or “for some”.\u00a0 It is denoted by\u00a0\u2203.<\/p>\n

      \n

      Examples<\/h3>\n

      Determine if each statement uses the universal or existential quantifier.<\/p>\n

        \n
      1. All mathematicians wear glasses.<\/li>\n
      2. There exists some number such that x2<\/sup> = x.<\/li>\n
      3. All politicians are dishonest.<\/li>\n<\/ol>\n
        Show Answer<\/span><\/p>\n
        1. Universal 2. Existential 3. Universal<\/div>\n<\/div>\n<\/div>\n

        Negating Quantified Statements<\/h3>\n

        Recall that negating a statement changes its truth value.\u00a0 So negating a true statement gives a false statement and negating a false statement gives a true statement.<\/p>\n

        \n

        Key Takeaways<\/h3>\n

        The negation of a universal statement (All A are B) is an existential statement (There exists some A that is not B).<\/p>\n

        The negation of an existential statement (There exists some A that is B) is a universal statement (All A are not B).<\/p>\n<\/div>\n

        \n

        Examples<\/h3>\n

        Negate each statement.<\/p>\n

          \n
        1. All mathematicians wear glasses.<\/li>\n
        2. There exists some number such that x2<\/sup> = x.<\/li>\n
        3. All politicians are dishonest.<\/li>\n<\/ol>\n

          1: There exists at least one mathematician that does not wear glasses.<\/p>\n

          This is a universal statement.\u00a0 So the negation will be an existential statement.\u00a0 Notice that the original statement was false, and the negation is true.<\/p>\n

          2: For all numbers, x2\u00a0<\/sup>\u2260 x.<\/p>\n

          This is an existential statement, so the negation will be a universal statement.\u00a0 Notice that the original statement is true (0 and 1 fit the property), and the negation is false.<\/p>\n

          3: There exists at least one politician that is honest.<\/p>\n

          This is a universal statement.\u00a0 So the negation will be an existential statement.<\/p>\n<\/div>\n

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