Logical Reasoning - Math Problem Example

Euler circles otherwise known as Euler diagrams are graphical representations of logical statements. Each set in this diagram is represented as a circle or closed curve. Intersections between sets are shown as partially overlapping circles. On the other hand, the equivalence of two sets is given by a complete overlap of the circles. Using this approach, the relationship of sets becomes clear. It is then easy to determine the validity of a conclusion based on the diagram. For the first statement it is clear from the diagram that given the premise, there is no established relation between B and C.

Likewise, the second statement also establishes no relationship between the two. Therefore, both statements are invalid. In contrast, truth tables are textual representations of the logic statement. Unlike a pure logic statement, the truth table presents all possible combinations of the input set and determines the conclusion from each combination. While not as intuitive as Euler circles, this approach provides a more exhaustive view of a logical statement. The respective truth tables for the above logical statements follow: In the first table, when all the premises are satisfied, there is definitely no way wherein the conclusion is satisfied.

This statement is therefore invalid. For the second statement, most cases satisfy the statement except for one. However, since the conclusion is an implication, then the presence of this single case invalidates the statement.Conclusion From the given examples, it is clear that the use of alternative representations of logical statements aid in logical reasoning. In fact, both Euler circles and truth tables were consistent in invalidating the logic for both examples.Question 1Let: Equations:Solution:Modify eqn 1: Substitute to eqn 2: Conclusion: UniquePhil has 10 $0.37 stamps.Question 2Solution: Using the Venn DiagramExactly 2 writing instruments: Conclusion: 15 backpacks contain exactly two of the three writing instruments.

Question 3Question 4Solution:Conclusion: The car got 54.4 miles per gallon.Question 5Question 6BiconditionalQuestion 72.8700Question 8Question 9Solution:Conclusion: cups of flour should be used.Question 10Solution:During peak hours: During off-peak hours: Conclusion: $1.76 can be saved by making 16-minute calls during off-peak hours.Question 11Solution:Using the Law of Contraposition: Using De Morgan's Law: Conclusion: They are logically equivalent.Question 12Solution: truth tablepqqpqTRUETRUEFALSEFALSETRUEFALSETRUETRUEFALSETRUEFALSEFALSEFALSEFALSETRUEFALSEConclusion: There is only 1 T in the final columb.

Question 13Question 14Question 15ValidQuestion 16Question 17Question 18InvalidQuestion 19Solution:1 3 7 13 15 19 25 27 31 37 ---Diff 2 4 6 2 4 6