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## MGF 1106

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**MGF 1106**Test 2 Review and Practice Solutions**You will need to have your own calculator for the test.**• You may not share calculators or use any type of communication device in place of a calculator. • Tests may not be made up for any reason other than a mandatory school – sponsored activity for which you must miss class. • If you miss one test for any other reason, your final exam score will be substituted for that test. A second missed test is a zero. No homework bonuses are awarded on a test when the final exam is substituted or you receive a zero on a missed test.**Be sure to complete each assignment with a score of at least**80% to receive the 10 point bonus.**1) Construct a truth table for the following statements.**a) p ∧ (~p ∨ q)**2) Determine the truth value for each statement when p is**true, q is true, and r is false. b) (p ∨ r)→ ~(p ∧ q) (T ∨ F)→ ~(T ∧ T) T→ ~T T→ F False c) ~[(p↔q) ∨ (q→~r)] ~[(T↔T) ∨ (T→~F)] ~[T ∨ (T→T)] ~[T ∨ T] ~T False a) ~p ∨ ~q ~T ∨ ~T F ∨ F False**3) Consider the statement “If you do a little bit each**day then you’ll get by, and if you do not do a little bit each day then you won’t (get by).” p: You do a little bit each day q: You get by The statement is true when both p and q have the same truth values. In other words, the statement is true when p and q are both true and when p and q are both false**4) Consider the statement (p ∧ q) ∧ (~p ∨ ~q).Is the**statement a tautology, self – contradiction, or neither a tautology nor a self - contradiction? The statement is a self – contradiction since all final truth values are false.**5) Consider the statements (p ∨ r)→ ~ q, (~p ∧ ~r**)→q.Are the statements equivalent? Not equivalent since the final truth values do not match up.**6) Write the converse, inverse, and contrapositive of each**statement. a) If today is Monday, then I do not have to go to school. Converse: If I do not have to go to school, then today is Monday. Inverse: If today is not Monday, then I have to go to school. Contrapositive: If I do have to go to school, then today is not Monday. b) If all birds fly south for the winter, then some bird-watchers are not happy. Converse: If some bird-watchers are not happy, then all birds fly south for the winter. Inverse: If some birds do not fly south for the winter, then all bird watchers are happy. Contrapositive: If all bird watchers are happy, then some birds did not fly south for the winter.**7) Negate the following statement.**If today is Monday, then I do not have to go to school. It is Monday and I do have to go to school.**8) Negate the following statements.**a) I practice the piano or I play outside. I don’t practice the piano and I don’t play outside. b) I apply myself and I don’t disappoint my parents. I don’t apply myself or I disappoint my parents.**9) Determine if each argument is valid or invalid.**a) If we sing loudly, then the neighbors will hear us. If the neighbors hear us, then the neighbors will smile. ∴ If the neighbors are smiling, then we sung loudly. Invalid: Misuse of Transitive b) If it is Thanksgiving, then I will not skip dessert. I skipped dessert. ∴ It is not Thanksgiving. Valid: Contrapositive. c) He is here or I am working from home. He is here. ∴ I am not working from home. Invalid: Misuse of disjunctive**10) Give a valid conclusion based on the premises.**a) If I am a full-time student, I cannot work. If I cannot work, I must budget my money carefully. I am a full time student. Therefore, …I must budget my money carefully. b) If a person is older than nine, then that person must pay the adult price to enter the Magic Kingdom. Becca did not pay the adult price to enter the Magic Kingdom. Therefore, …Becca is not older than nine. c) JoAnn is hungry or Bella is tired. JoAnn is not hungry. Therefore, …Bella is tired.**11) Use an Euler diagram to determine the validity of each**argument. Write “valid” or “invalid”. a) All poets appreciate language. All novelists appreciate language. Therefore, all poets are novelists. Invalid: b) All freshmen live on campus. No people who live on campus can own cars. Therefore, no freshman can own a car. Valid:**12) The measure of an angle is twenty-four more than twice**its complement. Find the measure of each angle.**13) Find the measure of ∠A in the triangle below. The**triangle is not necessarily drawn to scale.**14) A man who is 6 feet tall is standing 20 feet away from**a building. If the man’s shadow is ten feet long, how tall is the building?**15) A man is constructing a sail in the shape of a right**triangle. The diagonal side of the sail measures 39 meters. The shorter side of the sail measures 15 meters. How long is the other side of the sail?**16) A designer must create a decorative border around a**picture frame. The frame is five feet wide and seven feet long. The decorative border costs $12.75 per yard. How much will the border cost?**18) On “Decorator’s Challenge”, Tim was inspired to**create a circular patio. The patio has a diameter of 16 feet and Tim learned that his material would cost $5.75 per square foot. What is Tim’s cost to create his patio? Please round your answer to the nearest dollar. You may use either the “π” key on your calculator or 3.14 in calculating your answer.**19) Find the volume of a cargo container measuring 250 feet**by 150 feet by 30 feet.**20) Work problem 35 on page 582.**To find the distance across a lake, a surveyor took the following measurements. How far is it across the lake? 529 yards**MGF 1106**Extra Practice Problems**1) Construct a truth table for the following statement.**• ~(p ∨~ q)**1) Construct a truth table for the following statement.**• ~(p ∨~ q)**2) Determine the truth value for the statement when p is**false, q is true, and r is false. ~p↔(~q ∧r)**2) Determine the truth value for the statement when p is**false, q is true, and r is false. ~p↔(~q ∧r) ~False↔(~True ∧ False) True↔(False ∧ False) True ↔False False**3) Consider the statement “You’re blushing or**sunburned, and you’re not sunburned. When is the statement true? p: You’re blushing q: You’re sunburned.**3) Consider the statement “You’re blushing or**sunburned, and you’re not sunburned. When is the statement true? p: You’re blushing q: You’re sunburned. The statement is true when p is true and q is false.**4) Consider the statement Is the statement a tautology,**self – contradiction, or neither a tautology nor a self - contradiction?**4) Consider the statement Is the statement a tautology,**self – contradiction, or neither a tautology nor a self - contradiction? The statement is a tautology since all final truth values are true.**5) Consider the statements ~ p → (q ∨ ~r), (r∧~q**)→pAre the statements equivalent?**5) Consider the statements ~ p → (q ∨ ~r), (r∧~q**)→pAre the statements equivalent? Equivalent since the final truth values match up.**6) Write the converse, inverse, and contrapositive of the**statement. If the review session is successful, then no students fail the test.**6) Write the converse, inverse, and contrapositive of the**statement. If the review session is successful, then no students fail the test. Converse: If no students fail the test, then the review session is successful. Inverse: If the review session is not successful, then some students fail the test. Contrapositive: If some students fail the test, then the review session is (was) not successful.**7) Negate the following statement.**If there is a tax cut, then all people have extra spending money.**7) Negate the following statement.**If there is a tax cut, then all people have extra spending money. There is a tax cut and some people do not have extra spending money.**8) Negate the following statement.**They see the show and they do not have tickets.**8) Negate the following statement.**They see the show and they do not have tickets. They do not see the show or they have tickets.**9) Determine if the argument is valid or invalid.**If the defendants DNA is found at the crime scene, then we can have him stand trial. He is standing trial. ∴ We found evidence of his DNA at the crime scene.**9) Determine if the argument is valid or invalid.**If the defendants DNA is found at the crime scene, then we can have him stand trial. He is standing trial. ∴ We found evidence of his DNA at the crime scene. Invalid: Fallacy of the converse**10) Give a valid conclusion based on the premises.**If all students get requirements out of the way early, then no students take required courses in their last semester. Some students take required courses in their last semester.**10) Give a valid conclusion based on the premises.**If all students get requirements out of the way early, then no students take required courses in their last semester. Some students take required courses in their last semester. Therefore, … some students do not get requirements of their way early.**11) Use an Euler diagram to determine the validity of each**argument. Write “valid” or “invalid”. All people who arrive late cannot perform. All people who cannot perform are ineligible for scholarships. Therefore, all people who arrive late are ineligible for scholarships.**11) Use an Euler diagram to determine the validity of each**argument. Write “valid” or “invalid”. All people who arrive late cannot perform. All people who cannot perform are ineligible for scholarships. Therefore, all people who arrive late are ineligible for scholarships. ineligible Valid Can’t perform late**12) The measure of an angle is three less than two times**its supplement. Find the measure of each angle.