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Mth_151_Chapter3_handout

# Mth_151_Chapter3_handout - MTH 151 Quantifiers Part I...

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MTH 151 - Quantifiers Part I - Rewrite each statement using another quantifier without changing the meaning of the original statement. 1. There is at least one math student who loves logic. 2. No student doesn't love math. 3. Some days are not sunny. 4. There exists a rational number that is a real number. 5. All students are not math majors. 6. Not all integers are even. Part II - Write the negation of each statement. 1. Some people do not pay taxes. 2. All counting numbers are divisible by 1. 3. Some rectangles are squares. 4. No even integers are divisible by five. 5. At least one integer is negative. 6. Every counting number is a positive integer.

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E ULER D IAGRAMS AND A RGUMENTS W ITH Q UANTIFIERS PURPOSE: The purpose of this activity is to develop the ability to use Euler diagrams to test the validity of an argument when one or more statement contains a quantifier. REQUISITES: This activity assumes that students have been introduced to the use of Euler diagrams to represent statements with universal and/or existential quantifiers. DEFINITION: In order for an argument to be valid, the conclusion must always be true whenever the premises are true. An argument that is not valid is invalid. DIRECTIONS: Determine whether the following arguments (syllogisms) are valid by constructing an Euler diagram in which the premises hold. If the diagram shows the conclusion, without ambiguity, then the argument is valid. Otherwise, the argument is invalid. Example: Some students are lazy. All males are lazy. Some students are males. Consider the following Euler diagram: Notice that both premises hold, but the conclusion is not shown in the diagram. Notice that the ovals for “students” and “lazy people” overlap which matches the first premise. Notice that the oval for “males” is entirely within the oval for “lazy people” which matches the second premise. Now consider another possible Euler diagram: In the above diagram, the premises hold and. the conclusion is shown. However, in order for an argument to be valid, the conclusion must always be true whenever the premises are true. The premises say nothing about the relation between “students” and “males” so both diagrams must be used. To determine the validity of an argument we must consider all cases. Since the first diagram gives a case where the conclusion is not shown, even though the premises hold, the argument is invalid. Euler diagrams are drawn from the premises only. The conclusion is tested in all cases presented by the Euler diagrams.
Construct a Euler diagram for each of the following arguments and determine the validity. 1. All students are lazy. Nobody who is wealthy is a student . Lazy people are not wealthy.

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Mth_151_Chapter3_handout - MTH 151 Quantifiers Part I...

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