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Unformatted text preview: Math 109 Midterm Exam 1 Solution 1. (6 points) Let D be a division ring. Consider the following statement: D is a field is a necessary condition for D to be finite. For purposes of this question, it is not necessary to know what division ring, field, or finite mean. (a) Write the contrapositive of the statement. The statement is logically equivalent to the statement: if D is finite, then D is a field. Thus, its contrapositive is: if D is not a field, then D is not finite. (Note: the contrapositive could also be written: D is not finite is a necessary condition for D to not be finite.) (b) Write the converse of the statement. The converse of the statement is: if D is a field, then D is finite. (c) Write the negation of the statement. The statement is logically equivalent to the disjunction: D is not finite or D is a field. Thus, its negation is: D is finite and D is not a field. (Note: Perhaps it would be more precise to write the statement as a universally quantified statement as follows: For every division ring...
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- Spring '06