Solutions Lesson on Sentences and Negation negation_ex1.jpg Each of these sentences is a closed sentence (an objective statement which is either true or false). ___________________________________________________________________________________________________ negation_ex2.jpg "Jenny does not ride the bus" is the negation of "Jenny rides the bus." The negation of p is "not p." ___________________________________________________________________________________________________ negation_ex3.jpg The statement ~x represents the negation of x. ___________________________________________________________________________________________________ negation_ex4.jpg The statement ~a represents the negation of a. ___________________________________________________________________________________________________ negation_ex5.jpg Choice 4 has a variable in it. An open sentence is a statement which contains a variable and becomes either true or false depending on the value that replaces the variable. ___________________________________________________________________________________________________ Lesson on Conjunction conjunction_ex1.jpg "Jill eats pizza and Sam eats pretzels" is a conjunction. A conjunction is a compound statement formed by joining two statements with the connector AND. ___________________________________________________________________________________________________ conjunction_ex2.jpg The conjunction "p and q" is symbolized by p and.gif q. ___________________________________________________________________________________________________ conjunction_ex3.jpg A conjunction is a compound statement formed by joining two statements with the connector AND. ___________________________________________________________________________________________________ conjunction_ex4.jpg The truth value of a and.gif b is false. A conjunction is true when both of its combined parts are true, otherwise it is false. ___________________________________________________________________________________________________ conjunction_ex5.jpg When y = 2, the statement r is true and the statement s is true (i.e., The number 2 is both prime and even). Therefore, the conjunction r and.gif s is true when y = 2. ___________________________________________________________________________________________________ Lesson on Disjunction disjunction_ex1.jpg A disjunction is a compound statement formed by joining two statements with the connector OR. ___________________________________________________________________________________________________ disjunction_ex2.jpg The statement x or.gif y is a disjunction. ___________________________________________________________________________________________________ disjunction_ex3.jpg A disjunction is a compound statement formed by joining two statements with the connector OR. ___________________________________________________________________________________________________ disjunction_ex4.jpg If b is true then ~b is false. A disjunction is false when both statements are false. Therefore, the disjunction a or.gif ~b is false. ___________________________________________________________________________________________________ disjunction_ex5.jpg When y = 3, the statement r is true and the statement s is false. Therefore, all three choices list true statements. ___________________________________________________________________________________________________ Lesson on Conditional Statements conditional_ex1.jpg A conditional statement is an if-then statement in which p is a hypothesis and q is a conclusion. ___________________________________________________________________________________________________ conditional_ex2.jpg The hypothesis is r and the conclusion is s. The logical connector in a conditional statement is denoted by the symbol conditional.gif . ___________________________________________________________________________________________________ conditional_ex3.jpg The conditional is defined to be true unless a true hypothesis leads to a false conclusion. ___________________________________________________________________________________________________ conditional_ex4.jpg When x=2, hypothesis a is true and conclusion b is false. When a true hypothesis leads to a false conclusion, the conditional is false. Thus when x=2, conditional a conditional.gif b is false. ___________________________________________________________________________________________________ conditional_ex5.jpg When x=9, hypothesis a is false and conclusion b is true. By definition, conditional a conditional.gif b is true. ___________________________________________________________________________________________________ Lesson on Compound Statements compound_ex1.jpg The compound statement (a or.gif b) conditional.gif ~b is a conditional, where the hypothesis is the disjunction "a or b" and the conclusion is ~b. ___________________________________________________________________________________________________ compound_ex2.jpg If r and s are false statements, then (~r and.gif s) conditional.gif s is true as shown in the truth table below. r s ~r ~r and.gif s (~r and.gif s) conditional.gif s F F T F T ___________________________________________________________________________________________________ compound_ex4.jpg The truth values of (~x or.gif y) conditional.gif y are shown in the truth table below. x y ~x ~x or.gif y (~x or.gif y) conditional.gif y T T F T T T F F F T F T T T T F F T T F ___________________________________________________________________________________________________ compound_ex5.jpg The truth values of ~p conditional.gif (p and.gif ~q) are {T, T, F, F}, as shown in the truth table below. p q ~p ~q p and.gif ~q ~p conditional.gif (p and.gif ~q) T T F F F T T F F T T T F T T F F F F F T T F F ___________________________________________________________________________________________________ Lesson on Biconditional Statements biconditional_ex1.jpg Biconditional p biconditional_transp.gif q represents "p if and only if q," where p is a hypothesis and q is a conclusion. ___________________________________________________________________________________________________ biconditional_ex2.jpg The hypothesis is "11 is prime" and the conclusion is "11 is odd". So r biconditional_transp.gif s represents, "11 is prime if and only 11 is odd." The "if and only if" is abbreviated with "iff" in choice 3. ___________________________________________________________________________________________________ biconditional_ex3.jpg When proving the statement p iff q, it is equivalent to proving both of the statements "if p, then q" and "if q, then p". Since these conditionals were given in the problem, x biconditional_transp.gif y is biconditional. Therefore, each statement listed in choice 1, 2 and 3 is true. ___________________________________________________________________________________________________ biconditional_ex4.jpg The biconditional p biconditional_transp.gif q represents "p if and only if q", where p is a hypothesis and q is a conclusion. So m is the hypothesis of m biconditional_transp.gif n. ___________________________________________________________________________________________________ biconditional_ex5.jpg None of these statements is biconditional: one can sleep without snoring; Mary can eat pudding today that is not custard; it can be cloudy without any rain. ___________________________________________________________________________________________________ Lesson on Tautologies tautologies_ex1.jpg A compound statement, that is always true regardless of the truth value of the individual statements, is defined to be a tautology. The disjunction of a statement and its negation is a tautology. ___________________________________________________________________________________________________ tautologies_ex2.jpg No, the conditional statement s conditional.gif ~s is not a tautology. See the truth table below. s ~s s conditional.gif ~s T F F F T T ___________________________________________________________________________________________________ tautologies_ex3.jpg Yes, the statement [(p or.gif q) and.gif ~p] conditional.gif q is a tautology since its truth values are {T, T, T, T} as shown in the truth table below. p q ~p p or.gif q (p or.gif q) and.gif ~p [(p or.gif q) and.gif ~p] conditional.gif q T T F T F T T F F T F T F T T T T T F F T F F T ___________________________________________________________________________________________________ tautologies_ex4.jpg Yes, the statement ~(x or.gif y) biconditional_transp.gif (~x and.gif ~y) is a tautology since its truth values are {T, T, T, T} as shown in the truth table below. x y ~x ~y x or.gif y ~(x or.gif y) ~x and.gif ~y ~(x or.gif y) biconditional_transp.gif (~x and.gif ~y) T T F F T F F T T F F T T F F T F T T F T F F T F F T T F T T T ___________________________________________________________________________________________________ tautologies_ex5.jpg A conjunction is true when both parts are true. Since a statement and its negation have opposite truth values, the conjunction of a statement and its negation could never be true. ___________________________________________________________________________________________________ Lesson on Equivalence equivalence_ex1.jpg The truth values of (p and.gif ~q) conditional.gif ~p are {T, F, T, T} as shown in the truth table below. p q ~p ~q p and.gif ~q (p and.gif ~q) conditional.gif ~p T T F F F T T F F T T F F T T F F T F F T T F T ___________________________________________________________________________________________________ equivalence_ex2.jpg The statement p conditional.gif q is logically equivalent to the statement (p and.gif ~q) conditional.gif ~p, since they both have the same truth values, as shown in the truth table below. p q p conditional.gif q (p and.gif ~q) conditional.gif ~p T T T T T F F F F T T T F F T T ___________________________________________________________________________________________________ equivalence_ex3.jpg The statement q conditional.gif p is logically equivalent to the statement q conditional.gif (p and.gif q) since they both have the same truth values, as shown in the truth table below. p q p and.gif q q conditional.gif (p and.gif q) q conditional.gif p T T T T T T F F T T F T F F F F F F T T ___________________________________________________________________________________________________ equivalence_ex4.jpg The statement (a and.gif b) conditional.gif b is logically equivalent to the statement a conditional.gif (a or.gif b) since they both have the same truth values, as shown in the truth table below. a b a or.gif b a and.gif b (a and.gif b) conditional.gif b a conditional.gif (a or.gif b) T T T T T T T F T F T T F T T F T T F F F F T T ___________________________________________________________________________________________________ equivalence_ex5.jpg Equivalent statements have the same truth values. Therefore, x and y satisfy the definition of a biconditional.. Thus, the statements listed in choice 1 and choice 3 are true. The biconditional of two equivalent statements is a tautology. Therefore, the statement listed in choice 2 is true. ___________________________________________________________________________________________________ Practice Exercises practice_ex1.jpg This truth table shows the truth values for the negation of p, and for the conjunction, disjunction and conditional of statements p and q. ___________________________________________________________________________________________________ practice_ex2.jpg This truth table shows the truth values for the compound statement (p and.gif q) conditional.gif ~q. ___________________________________________________________________________________________________ practice_ex3.jpg This truth table shows the truth values of various compound statements involving x and y. ___________________________________________________________________________________________________ practice_ex4.jpg The conditional statements in problem 3 are x conditional.gif y and y conditional.gif x. ___________________________________________________________________________________________________ practice_ex5.jpg The biconditional statement from problem 3 is x biconditional_transp.gif y. ___________________________________________________________________________________________________ practice_ex6.jpg This truth table shows the truth values of various compound statements involving a and b. ___________________________________________________________________________________________________ practice_ex7.jpg The statement in the last column of the truth table in problem 6 is a tautology since all of its truth values are true. ___________________________________________________________________________________________________ practice_ex8.jpg The truth values for the last column are all true. Thus the statement (p conditional.gif ~q) biconditional_transp.gif [~(p and.gif q)] is a tautology. ___________________________________________________________________________________________________ practice_ex9.jpg The statements p conditional.gif ~q and ~(p and.gif q) have the same truth value. These statements are, therefore, logically equivalent. ___________________________________________________________________________________________________ practice_ex10.jpg The biconditional of two equivalent statements is a tautology.