Determine whether the statement is a tautology, self-contradiction, or neither. (,q ( p) ( ( q
> Height of a Tree At a given time of day, the ratio of the height of an object to the length of its shadow is the same for all objects. If a 3-ft stick in the ground casts a shadow of 1.2 ft, find the height of a tree that casts a shadow that is 15.36 ft.
> In Exercises 7–16, construct a truth table for the statement. ((p (q)
> The Youngest Triplet The Barr triplets have an annoying habit: Whenever a question is asked of the three of them, two tell the truth and the third lies. When I asked them which of them was born last, they replied as follows. Mary: Katie was born last. Ka
> At Puzzle Solve the following puzzle. The Joneses have four cats. The parents are Tiger and Boots, and the kittens are Sam and Sue. Each cat insists on eating out of its own bowl. To complicate matters, each cat will eat only its own brand of cat food. T
> Buying a Boat Four partners decide to share the cost of a boat equally. By bringing in an additional partner, they can reduce the cost to each of the five partners by $3000. What is the total cost of the boat?
> Satisfiability Problem Refer to the Recreational Mathematics box on page 119 and then solve the following satisfiability problem. Allen, Booker, Chris, and Dennis all were born in the same year—one in January, one in February, one in March, and one in Ap
> Construct a truth table for (a) ( p ( q )((r ( s ) (b) ( q ( (p ) ( (r ( s )
> In Exercises 71 and 72, construct truth tables for the symbolic statement. [ ( r ( (q )((] ( ( q((r)
> In Exercises 71 and 72, construct truth tables for the symbolic statement. [p ( (q ( ( r)] ((p ( ( q)
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. Job Interview Consider the statement “If your interv
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. A New Computer Your parents make the following stat
> In Exercises 23–26, show that the set has cardinality /by establishing a one-to-one correspondence between the set of counting numbers and the given set. {3, 9, 27, 81, 243, …}
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. If Muhundan did not receive the outstanding teacher
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. Muhundan did not receive the outstanding teacher awa
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. If Muhundan did not receive the outstanding teacher
> Suppose both of the following statements are false. p: Muhundan spoke at the teachers’ conference. q: Muhundan received the outstanding teacher award. Find the truth values of each compound statement. If Muhundan received the outstanding teacher award,
> Convert each of the following to a numeral in the base indicated. 2921 to base 8
> Use the graph to determine the truth value of each simple statement. Then determine the truth value of the compound statement. The following graph shows the number of credits in various categories needed by Jose to earn his Associate in Arts degree from
> Use the graph to determine the truth value of each simple statement. Then determine the truth value of the compound statement.The following graph shows the number of credits in various categories needed by Jose to earn his Associate in Arts degree from V
> Use the information provided about the moons for the planets Jupiter and Saturn to determine the truth values of the simple statements. Then determine the truth value of the compound statement. Moon Comparisons If Jupiter has 16 moons or Saturn does no
> Use the information provided about the moons for the planets Jupiter and Saturn to determine the truth values of the simple statements. Then determine the truth value of the compound statement. Moon Comparisons Phoebe has a larger diameter than Rhea if a
> Use the information provided about the moons for the planets Jupiter and Saturn to determine the truth values of the simple statements. Then determine the truth value of the compound statement. Moons of Saturn Titan may have water and Titan may have atmo
> In Exercises 23–26, show that the set has cardinality / by establishing a one-to-one correspondence between the set of counting numbers and the given set. {2, 4, 8, 16, 32, … }
> Use the information provided about the moons for the planets Jupiter and Saturn to determine the truth values of the simple statements. Then determine the truth value of the compound statement. Jupiter’s Moons Io has a diameter of 1000
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. Honda makes automobiles or Honda makes motorcycles, if and only if Mazda makes cereal.
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. Independence Day is in July and Labor Day is in September, if and only if Thanksgiving is in April.
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. Spike Lee is a movie director, or if Halle Berry is a school teacher then George Clooney is a circus clown.
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. Apple makes computers, if and only if Nike makes sports shoes or Rolex makes watches.
> X-rays With a certain medical insurance policy, the customer must first pay an annual $100 deductible, and then the policy covers 80% of the cost of x-rays. The first insurance claims for a specific year submitted by Yungchen are for two x-rays. The firs
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. If the moon circles the earth and the earth circles the sun, then the moon is made of cheese.
> Determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. If a cat has whiskers or a fish can swim, then a dog lays eggs.
> If p is true, q is false, and r is false, find the truth value of the statement. ([( p ( q) (( p ( ( r)]
> If p is true, q is false, and r is false, find the truth value of the statement. ((p ( r) ( (( q ( r)
> In Exercises 23–26, show that the set has cardinality / by establishing a one-to-one correspondence between the set of counting numbers and the given set. {1, 4, 9, 16, 25, p }
> If p is true, q is false, and r is false, find the truth value of the statement. ( [ p ( (q ( r)]
> If p is true, q is false, and r is false, find the truth value of the statement. ((p ((q) ( ( r
> If p is true, q is false, and r is false, find the truth value of the statement. r ( (( p ( ( q)
> If p is true, q is false, and r is false, find the truth value of the statement. (p ( q) ( (q ( ( r )
> If p is true, q is false, and r is false, find the truth value of the statement. ( p ( ( q) ( ( r
> If p is true, q is false, and r is false, find the truth value of the statement. p ( (q (r)
> Convert each of the following to a numeral in the base indicated. 1098 to base 8
> Determine whether the statement is an implication. [( p ( q) ( r] ( ( p ( q)
> Determine whether the statement is an implication. [( p (q) ( (q ( p)] (( p ( q)
> Determine whether the statement is an implication. ( p ( q) ( ( p ( ,r)
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets.
> Determine whether the statement is an implication. (p ( ( ( p ( q)
> Determine whether the statement is an implication. p ( ( p ( q)
> Determine whether the statement is an implication. ( p ( p
> Determine whether the statement is a tautology, self-contradiction, or neither. (( p ( q) ( (p ( ( q)
> Determine whether the statement is a tautology, self-contradiction, or neither. (p (( q) 4 ((p ( q)
> Determine whether the statement is a tautology, self-contradiction, or neither. (p ( (q ( ( q)
> Investing You place $1000 in a mutual fund. The first year, the value of the fund increases by 10%. The second year, the value of the fund decreases by 10%. Determine the value of the fund at the end of the second year. Is it greater than, less than, or
> Determine whether the statement is a tautology, self-contradiction, or neither. ((p ( q) ( (q
> Determine whether the statement is a tautology, self-contradiction, or neither. (p ( q) (( q
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets.
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. It is false that if Elaine went to lunch, then she cannot take a message and we will have to go home.
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. If it is not too cold then we can take a walk, or we can go to the gym.
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. If the dam holds then we can go fishing, if and only if the pole is not broken.
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. The election was fair if and only if the polling station stayed open until 8 p.m., or we will request a recount
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. We will advance in the tournament if and only if Max plays, or Pondo does not show up.
> Write the statement in symbolic form. Then construct a truth table for the symbolic statement. If today is Monday, then the library is open and we can study together.
> Construct a truth table for the statement. ((p ( (q) ( ((q ( r)
> Construct a truth table for the statement. ( p ( q) ( ((q ( (r)
> Convert each of the following to a numeral in the base indicated. 549 to base 7
> In Exercises 3–12, show that the set is infinite by placing it in a one-to-one correspondence with a proper subset of itself. Be sure to show the pairing of the general terms in the sets. { 2, 3, 4, 5, 6, …}
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets.
> Construct a truth table for the statement. [r ( (q ( ( p)] ( (p
> Construct a truth table for the statement. (( r ( (q) ( p
> Construct a truth table for the statement. ( p ( r) ( (q ( r)
> Construct a truth table for the statement. (p ( (q ( (r)
> Construct a truth table for the statement. ((p ( q) ( r
> Construct a truth table for the statement. (p((q ( r)
> Construct a truth table for the statement. ( p ( q) ( ( p ( q)
> construct a truth table for the statement. ( ( p ( ( q)
> Construct a truth table for the statement. ((q ( p) ( ( q
> Construct a truth table for the statement. p ( (q ( p)
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets.
> Adjusting for Inflation Assume that the rate of inflation is 2% per year for the next 2 years. What will the price of a sofa be 2 years from now if the sofa costs $999 today?
> Construct a truth table for the statement. ( p ( q)(p
> Construct a truth table for the statement. (p ( q) ( p
> Construct a truth table for the statement. ((p ( q)
> Construct a truth table for the statement. ( p ( ( q
> In Exercises 7–16, construct a truth table for the statement. (p ( (q
> Construct a truth table for the statement. (( p ( ( q )
> Construct a truth table for the statement. (p ( ( q
> Construct a truth table for the statement. p ( ( q
> Construct a truth table for the statement. q ( ( p
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets. {7, 11, 15, 19, 23, p }
> Construct a truth table for the statement. p ( ( p
> Convert each of the following to a numeral in the base indicated. 102 to base 5
> Construct a truth table for the statement. p ( ( p
> Must ( p ( ( q ) ( r and ( q ( ( r ) ( p have the same number of trues in their answer columns? Explain.
> On page 107, we indicated that a compound statement consisting of n simple statements had 2n distinct true–false cases. a) How many distinct true–false cases does a truth table containing simple statements p, q, r, and s have? b) List all possible true–f
> Construct a truth table for the symbolic statement. [ ( q ( ( r ) ( ( ( p ( ( q ) ] ( ( p ( ( r )
> Construct a truth table for the symbolic statement. ( [ ( ( ( p ( q ) ) ( ( q ( r) ]
> Airline Special Fares An airline advertisement states, “To get the special fare you must purchase your tickets between January 1 and February 15 and fly round trip between March 1 and April 1. You must depart on a Monday, Tuesday, or
> Read the requirements and each applicant’s qualifications for obtaining a loan. (a) Identify which of the applicants would qualify for the loan. (b) For the applicants who do not qualify for the loan, explain why. To qualify for a loan of $45,000, an ap
> Read the requirements and each applicant’s qualifications for obtaining a loan. (a) Identify which of the applicants would qualify for the loan. (b) For the applicants who do not qualify for the loan, explain why. To qualify for a loan of $40,000, an app
> In Exercises 13–22, show that the set has cardinal number / by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets. {2, 5, 8, 11, 14, p }
> In Exercises 3–12, show that the set is infinite by placing it in a one-to-one correspondence with a proper subset of itself. Be sure to show the pairing of the general terms in the sets.
