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Logic Problems

Before starting the axiomatic method in any area of mathematics certain basic logic principles and a well-defined starting point must be established. The following list of basic logic problems concerns constructing truth tables, working with quantifiers, and writing mathematical statements.

In exercises 1-20, let and be logical variables representing mathematical statements.A truth table is a table used to compute the functional values of logical expressions on each combination of values taken by their logical variables.

Problem 1. Define the logical connective And which is denoted by Also show a truth table for this definition.

Problem 2. Define the logical connective Or which is denoted by Also show a truth table for this definition.

Problem 3. Define the logical connective Not which is denoted by Also show a truth table for this definition.

Problem 4. Define the logical connective Implies which is denoted by Also show a truth table for this definition.

Problem 5. Define tautology and illustrate an example.

Problem 6. Define contradiction and illustrate an example.

Problem 7. Define contingency and illustrate an example.

Problem 8. Construct a truth table for and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 9. Construct a truth table for and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 10. Define contrapositive and illustrate an example.

Problem 11. Define converse and illustrate an example.

Problem 12. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 13. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 14. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 15. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 16. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 17. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 18. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 19. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

Problem 20. Construct a truth table for the statement and state which kind of proposition this is, a tautology, contradiction, or a contingency.

In exercises 21-30, let and represent the set of natural numbers and the set of integers, respectively.

Problem 21. Write out the statement using words rather than symbols.

Problem 22. Write out the statement using words rather than symbols.

Problem 23. Write out the statement using words rather than symbols.  

Problem 24. Write out the statement using words rather than symbols.

Problem 25. Write out the statement using words rather than symbols.

Problem 26. Write out the statement using words rather than symbols.  

Problem 27. Write out the statement using words rather than symbols.

Problem 28. Write out the statement using words rather than symbols.  

Problem 29. Write out the statement: "For all integers and , the numbers and are equal." using symbols rather than words.  

Problem 30. Write out the statement: "Given any real number there exists a natural number such that is strictly less than ." using symbols rather than words.  

Problem 31. Write out the statement: "Given any real number there exists a natural number such that the sum of and is zero." using symbols rather than words.  

Problem 32. Write out the statement: "Given any nonnegative real number there exists a natural number such that " using symbols rather than words.  

Problem 33. Write out the statement: "Given any nonzero real number thee exists a natural number such that " using symbols rather than words.  

Problem 34. Write out the statement: "There exists a smallest natural number" using symbols rather than words.  

Problem 35. Write out the statement: "There is no largest integer." using symbols rather than words.  

Problem 36. Write out the statement: "Given any two distinct real numbers, some rational number lies strictly between them. " using symbols rather than words.  

Problem 37. Write out a meaningful negation of the statement in problem 30 using symbols and then words. Speculate whether the negation statement is true or false using either the original statement or the negation of the statement.

Problem 38. Write out a meaningful negation of the statement in problem 31 using symbols and then words. Speculate whether the negation statement is true or false using either the original statement or the negation of the statement.

Problem 39. Write out a meaningful negation of the statement in problem 34 using symbols and then words. Speculate whether the negation statement is true or false using either the original statement or the negation of the statement.

Problem 40. Write out a meaningful negation of the statement in problem 36 using symbols and then words. Speculate whether the negation statement is true or false using either the original statement or the negation of the statement.

Logic Problems Books