Math Logic Problems

A list of basic math logic.

In exercises 1-20, let $math logic problems _gr_1.gif]$ and $math logic problems _gr_2.gif]$ 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.

(1) Define the logical connective And which is denoted by $math logic problems _gr_3.gif]$ Also show a truth table for this definition.

(2) Define the logical connective Or which is denoted by $math logic problems _gr_4.gif]$ Also show a truth table for this definition.

(3) Define the logical connective Not which is denoted by $math logic problems _gr_5.gif]$ Also show a truth table for this definition.

(4) Define the logical connective Implies which is denoted by $math logic problems _gr_6.gif]$ Also show a truth table for this definition.

(5) Define tautology.

(6) Define contradiction.

(7) Define contingency.

(8) Construct a truth table for $math logic problems _gr_7.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(9) Construct a truth table for $math logic problems _gr_8.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(10) Define contrapositive.

(11) Define converse.

(12) Construct a truth table for the statement $math logic problems _gr_9.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(13) Construct a truth table for the statement $math logic problems _gr_10.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(14) Construct a truth table for the statement $math logic problems _gr_11.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(15) Construct a truth table for the statement $math logic problems _gr_12.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(16) Construct a truth table for the statement $math logic problems _gr_13.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(17) Construct a truth table for the statement $math logic problems _gr_14.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(18) Construct a truth table for the statement $math logic problems _gr_15.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(19) Construct a truth table for the statement $math logic problems _gr_16.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(20) Construct a truth table for the statement $math logic problems _gr_17.gif]$ and state which kind of proposition this is, a tautology, contradiction, or a contingency.

(21) Define the symbol $math logic problems _gr_18.gif]$

(22) Define the symbol $math logic problems _gr_19.gif]$

In exercises 23-30, let $math logic problems _gr_20.gif]$ and $math logic problems _gr_21.gif]$ represent the set of natural numbers and the set of integers, respectively.

(23) Write out the statement $math logic problems _gr_22.gif]$ using words rather than symbols.

(24) Write out the statement $math logic problems _gr_23.gif]$ using words rather than symbols.

(25) Write out the statement $math logic problems _gr_24.gif]$ using words rather than symbols.

(26) Write out the statement $math logic problems _gr_25.gif]$ using words rather than symbols.

(27) Write out the statement $math logic problems _gr_26.gif]$ using words rather than symbols.

(28) Write out the statement $math logic problems _gr_27.gif]$ using words rather than symbols.

(29) Write out the statement $math logic problems _gr_28.gif]$ using words rather than symbols.

(30) Write out the statement $math logic problems _gr_29.gif]$ using words rather than symbols.

(31) Write out the statement: "For all integers $math logic problems _gr_30.gif]$ and $math logic problems _gr_31.gif]$ , the numbers $math logic problems _gr_32.gif]$ and $math logic problems _gr_33.gif]$ are equal." using symbols rather than words.

(32) Write out the statement: "Given any real number $math logic problems _gr_34.gif]$ there exists a natural number $math logic problems _gr_35.gif]$ such that $math logic problems _gr_36.gif]$ " using symbols rather than words.

(33) Write out the statement: "Given any real number $math logic problems _gr_37.gif]$ there exists a natural number $math logic problems _gr_38.gif]$ such that $math logic problems _gr_39.gif]$ " using symbols rather than words.

(34) Write out the statement: "Given any nonnegative real number $math logic problems _gr_40.gif]$ there exists a natural number $math logic problems _gr_41.gif]$ such that $math logic problems _gr_42.gif]$ " using symbols rather than words.

(35) Write out the statement: "Given any nonzero real number $math logic problems _gr_43.gif]$ thee exists a natural number $math logic problems _gr_44.gif]$ such that $math logic problems _gr_45.gif]$ " using symbols rather than words.

(36) Write out the statement: "There exists a smallest natural number" using symbols rather than words.

(37) Write out the statement: "There is no largest integer." using symbols rather than words.

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

(39) Write out the statement: "Given any positive real number $math logic problems _gr_46.gif]$ there exists a natural number $math logic problems _gr_47.gif]$ such that $math logic problems _gr_48.gif]$ whenever $math logic problems _gr_49.gif]$ is a natural number greater than $math logic problems _gr_50.gif]$ " using symbols rather than words.

(40) Write out the statement: "For each real number $math logic problems _gr_51.gif]$ if $math logic problems _gr_52.gif]$ then there exists a positive real number $math logic problems _gr_53.gif]$ such that for each number $math logic problems _gr_54.gif]$ if $math logic problems _gr_55.gif]$ then $math logic problems _gr_56.gif]$ " using symbols rather than words.

(41) Write out the negation of the statements in 16 through 20. Speculate whether each statement is true or false using either the original statement or the negation of the statement.

(42) Write out the negation of the statements in 31 through 35. Speculate whether each statement is true or false using either the original statement or the negation of the statement.

(43) Write out the negation of the statements in 36 through 40. Speculate whether each statement is true or false using either the original statement or the negation of the statement.

Cite this as:
Math Logic Problems
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/math-logic-problems.html