Exam 3: Logic

Determine which, if any, of the three statements are equivalent. -I. A check was mailed to you on Friday, or the check was not mailed to you on Thursday but the Check arrived late. II. If the check was not mailed to you on Friday, then the check was not mailed to you on Thursday but the check arrived late. III. The check arrived late, if and only if it was mailed to you on Friday or it was not mailed to you On Thursday.

Use the information given in the chart or graph to determine the truth values of the simple statements. Then determine the truth value of the compound statement given. - Planet X \bullet Moon 1 o Moon 2 Diameter of moons: May have: o Moon 3 \circ4-7 o water ice \circ Moon 4 \circ8-14 \bullet atmosphere \circ Moon 5 \bullet15-23 \circ both If Moon 4 has a diameter of 8-14 km or Planet X has four moons, then Moon 1 may have water ice.

Write an equivalent sentence for the statement. -If it is raining, you take your hat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.)

Write the compound statement in words. -Let $\mathrm { p } =$ "The monitor is included." $\mathrm { q } =$ "The color printer is optional." $r =$ "The zip drive is extra." $\sim ( p \wedge q ) \rightarrow ( r \vee \sim q )$

Use truth tables to test the validity of the argument. - (p\rightarrowq)\wedge(q-r) \therefore

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If a line does not intersect a circle in more than one point; then the line is tangent to the circle.

Use DeMorganʹs laws or a truth table to determine whether the two statements are equivalent. - $\mathrm { p } \vee \mathrm { q } , \sim ( \sim \mathrm { p } \wedge \sim \mathrm { q } )$

Given p is true, q is true, and r is false, find the truth value of the statement. - $r \rightarrow ( \sim p \vee q )$

Given p is true, q is true, and r is false, find the truth value of the statement. - $( \sim p \leftrightarrow \sim q ) \wedge ( p \leftrightarrow \sim r )$

Given p is true, q is true, and r is false, find the truth value of the statement. - $[ ( \sim \mathrm { p } \rightarrow \mathrm { r } ) \wedge ( \sim \mathrm { p } \vee \mathrm { q } ) ] \rightarrow \mathrm { r }$

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. - $7 < 10 \text { or } 1 > 4 \text {, and } 6 < 10$

Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. -8 × 2 = 20 if and only if 7 + 6 = 13.

Write an equivalent sentence for the statement. -It is false that cats are lazy or dogs are not friendly. (Hint: Use De Morganʹs laws.)

Write the compound statement in words. -Let $\mathrm { r } =$ "The puppy is trained." $\mathrm { p } =$ "The puppy behaves well." $q =$ "His owners are happy." $r \rightarrow ( p \wedge q )$

Write a negation of the statement. -Not all people like football.

Write the compound statement in symbols. Let r = ʺThe food is good,ʺ p = ʺI eat too much,ʺ q = ʺIʹll exercise.ʺ -Iʹll exercise if I eat too much.

Write a negation of the statement. -No fifth graders play soccer.

Determine whether the statement is a self-contradiction, an implication, a tautology (that is not also an implication), or none of these. - $( q \wedge p ) \leftrightarrow \sim ( p \wedge q )$

Determine the truth value for the simple statement. Then use these truth values to determine the truth value of the compound statement. Use the chart or graph when provided. -In Region X, people are spending more on leisure activities. Annual per capita spending in dollars: 2000 2010 Movies 38 186 Dining Out 95 492 Vacations 502 1786 Sporting Goods 149 343 Per capita spending on vacations was higher in Region X in 2010 than it was in 2000 or per capita spending on dining out was less in in Region X in 2010 than it was in 2000.

Construct a truth table for the statement. - $\sim(p \wedge q) \rightarrow \sim(p \vee q)$