Chapter 7 - Additional Exercises to be Done in Class 1. Symbolize the following sentences in PL using the given symbolization key. UD: Alfy, Barbara, Clarence, Dawn, Ellis, and the cities Houston, Indianapolis, Kalamazoo, Newark, Philadelphia, San Francisco, and Tulsa Bxy: x was born in y Lxy: x lives in y Axy: x is larger than y Txy: x is taller than y a: Alfy b: Barbara c: Clarence d: Dawn e: Ellis h: Houston i: Indianapolis k: Kalamazoo n: Newark p: Philadelphia s: San Francisco t: Tulsa a. Alfy was born in Indianapolis. b. Clarence was born in Tulsa. c. Barbara was born in Newark. d. Dawn was born in San Francisco. e. Ellis was born in Houston. f. No one was born in Kalamazoo. g. Philadelphia is larger than Houston, Houston is larger than Newark, and Newark is larger than Kalamazoo. h. Tulsa isn't larger than either Philadelphia or Houston. i. Indianapolis is larger than Houston if and only if it is larger than Philadelphia. j. Barbara lives in Philadelphia only if Dawn does. k. Everyone lives in Philadelphia, but no one was born there. 1. Barbara is taller than Clarence and Clarence is taller than Alfy, but neither Barbara nor Clarence is taller than Ellis. m. Dawn is the tallest person in the office. n. Alfy isn't taller than everyone else in the office. o. Alfy isn't taller than anyone in the office, but he is larger than everyone else in the office. p. If Clarence is taller than Barbara, he's also larger than Alfy. 1
2. Symbolize the following sentences in PL using the given symbolization key. UD: Andrea, Bentley, Charles, and Deirdre Bx: x is beautiful Ix: x is intelligent Rx: x is rich Axy: x is attracted to y Dxy: x despises y Lxy: x loves y Sxy: x is shorter than y a: Andrea b: Bentley c: Charles d: Deirdre a. Andrea is both intelligent and beautiful, but she is not rich. b. Charles is rich and beautiful but not intelligent. c. Deirdre is beautiful, rich, and intelligent. d. Bentley is neither rich, nor beautiful, nor intelligent, e. If Bentley is intelligent, so are both Deirdre and Andrea. f. Andrea is beautiful and intelligent, Bentley is intelligent but not beautiful, and neither is rich. g. Andrea loves Bentley but despises Charles. h. Andrea loves both herself and Charles and despises both Bentley and Deirdre. i. Charles neither loves nor despises Andrea but both loves and despises Deirdre. j. Neither Deirdre nor Bentley is attracted to Charles, but Charles is attracted to both of them. k. Charles is attracted to Bentley if and only if Bentley both is shorter than Charles and is rich. l. Andrea is attracted to both Bentley and Deirdre but doesn't love either of them. m. If Deirdre is shorter than Charles and Charles is shorter than Andrea, then Deirdre is shorter than Andrea. n. If Bentley is attracted to Deirdre and she is attracted to him, then they love each other. o. If Charles loves Bentley and Bentley loves Andrea, then Charles both despises and is shorter than Andrea. p. If Charles is neither rich nor beautiful nor intelligent, then no one loves him. q. Only Deirdre is rich. r. Only Deirdre is both rich and intelligent. 2
3. Symbolize the following sentences in PL using the given symbolization key. UD: The jellybeans in the jar on the coffee table Bx: x is black Rx: x is red a. All the jellybeans are black. b. Some of the jellybeans are black. c. None of the jellybeans is black. d. Some of the jellybeans are not black. e. Some of the jellybeans are black and some are red. f. If all the jellybeans are black then none is red. g. If some are red some are black. h. If none is black all are red. i. All are black if and only if none is red. j. Either all are black or all are red. 4. Symbolize the following sentences in PL. {Note: Not all of these sentences are true.) UD: The integers 1-100 Ex: x is even Ox: x is odd Lxy: x is less than y Px: x is prime Cix: x is greater than 0 a: 1 b: 2 c: 4 d: 100 a. Some integers are odd and some are even. b. Some integers are both odd and even. c. No integer is less than 1. d. No integer is greater than 100. e. Every integer is greater than 0. f. 100 is greater than every odd integer. g. There is a prime that is even. h. Some primes are not even. i. All prime integers greater than 2 are odd. j. 1 is not prime and is not greater than any integer. k. There is an integer that is greater than 2 and less than 4. 3
5. Symbolize the following sentences in PL using the given symbolization key. UD: The students in a logic class Px: x will pass Sx: x will study j: Jamie r: Rhoda a. If Jamie will pass everyone will pass. b. Either no one will pass or Jamie will pass. c. If anyone passes both Jamie and Rhoda will. d. Not everyone will pass, but Rhoda will. e. If Rhoda doesn't pass no one will. f. Some, but not all, of the students will pass. g. Rhoda will pass if Jamie does, and if Rhoda passes everyone will pass h. No one will pass if Jamie doesn't pass, and if she does everyone will. i. Everyone will study but not everyone will pass. j. If everyone studies everyone will pass. k. If everyone studies some will pass. 6. Which of the following are formulas of PL? (Here we allow the deletion of outer parentheses and the use of square brackets in place of parentheses.) For those that are not, explain why they are not. For those that are, state whether they are sentences or open sentences. a. Ba & Zz b. (x)px v Py c. ( y) ~ Hyy & Ga d. ( z) (Ex)(Fzx & Fxz) e. ( z)(( x)fzx & Fxz) f. ( x)faa g. ( z)(fz & Bgz) ( z)gzb h. ( x)[fx& ( x)(px Gx)] i. (~ x)(fx v Gx) j. ~ ( x)(gx ( z)fzx) k. ( x)( y)lxx 1. ( x)[( y)fyx ( y)fxy] m. (Bu & ~ Faa) ( w) ~ Fww n. ( a)fa o. Fw ( w)gww p. ( z)(hza ( z)gaz) 4
7. For each of the following formulas, indicate whether it is a sentence of PL. If it is not a sentence, explain why it is not. Also list all its subformulas, identifying the main logical operator of each. a. ( x)( y)byx b. ( x) ~ ( y)byx c. ( x) (~ Fx & Gx) (Bg Fx) d. ( y)[( z) ~ Byz v Byy] e. ~ ( x)px & Rab f. Rax ~ ( y)ryx g. ~ [~ ( x)fx ( w) ~ Gw] Maa h. ( x)( y)( z)mxyz & ( z)( x)( y)m)yzx i. ~~~ ( x)( z)(gxaz v ~ Hazb) j. ( z) [Fz ( w) (~ Fw & Gwaz) ] k. ( x)[fx ( w)(~ Gx ~ Hwx)] 1. ~ [( x)fx v ( x) ~ Fx] m. (Hb v Fa) ( z)(~ Fz & Gza) n. ( w)(fw & ~ Fw) (He & ~ He) 8. Indicate, for each of the following sentences, whether it is an atomic sentence, a truthfunctional compound, or a quantified sentence. a. ( x) (Fx Ga) b. ( x) ~ (Fx Ga) c. ~ ( x)(fx Ga) d. ( w)raw v ( w)rwa e. ~ ( x)hx f. Habc g. ( x)(fx ( w)gw) h. ( x)fx ( w)gw i. ( w)(pw ( y)(hy ~Kyw)) j. ~ ( w) (Jw v Nw) v ( w) (Mw v Lw) k. ~ [( w)(jw v Nw) v ( w)(mw v Lw)] 1. Da m. ( z)gza ( z)fz n. ~ ( x) (Fx & ~ Gxa) o. ( z) ~ Hza p. ( w)(~ Hw ( y)gwy) q. ( x) ~ Fx ( z) ~ Hza 5
9. For each of the following sentences, give the substitution instance in which 'a' is the instantiating term. a. ( w) (Mww & Fw) b. ( y)(mby Mya) c. ( z) ~ (Cz ~ Cz) d. ( x) [(Laa & Lab) Lax] e. ( z)[fz & ~ Gb) (Bzb v Bbz)] f. ( w)[fw & ( y)(cyw Cwa)] g. ( y) [~ ( z)nyz ( w)(mww & Nyw)] h. ( y)[(fy & Hy) [( z)(fz & Gz) Gy]] i. ( x)(fxb Gbx) j. ( x)( y)[( z)hzx ( z)hzy] k. ( x) ~ ( y)(hxy & Hyx) 1. ( z) [Fz ( w) (~ Fw & Gwaz) ] m. ( w)( y)[(hwy & Hyw) ( z)gzw] n. ( z) ( w) ( y) [ (Fzwy Fwzy) Fyzw] 10. Which of the following examples arc substitution instances of the sentence '( w)( y)(rwy Byy)'? a. ( y)ray Byy b. ( y) (Ray Byy) c. ( y) (Rwy Byy) d. ( y) (Rcy Byy) e. ( y) (Ryy Byy) f. ( y)(ray Byy) g. (Ray Byy) h. ( y) (Ray Baa) i. Rab Bbb 11. Which of the following examples are substitution instances of the sentence '( x)[( y) ~ Rxy Pxa]'? a. ( y) ~ Ray Paa b. ( y) ~ Raa Paa c. ( y) ~ Ray Pba d. ( y) ~ Rpy Ppa e. ( y) (~ Ryy Paa) f. ( y) ~ Ray Pya g. ( y) ~ Raw Paa h. ( y) ~ Rcy Pca 6
12. Identify each of the following sentences as either an A-, E-, I-, or O-sentence and symbolize each in PL using the given symbolization key. UD: A pile of coins consisting of quarters, dimes, nickels, and pennies Qz: z is a quarter Dz: z is a dime Nz: z is a nickel Cz: z contains copper Pz: z is a penny Sz: z contains silver Kz: z contains nickel Zz: z contains zinc Bz: z is a buffalo head coin Iz: z is an Indian head coin Mz: z was minted before 1965 a. All the pennies contain copper. b. Some of the dimes contain silver. c. Some of the dimes do not contain silver. d. None of the quarters contains silver. e. Some of the nickels are buffalo heads. f. All the nickels contain nickel. g. No penny contains silver. h. Some of the nickels are not buffalo heads. i. Every penny was mimed before 1965. j. Some quarters were not minted before 1965. k. Every coin containing silver contains copper. 1. No penny contains nickel, m. No coin that contains nickel contains silver. n. Every coin minted during or alter 1965 contains zinc. o. None of the quarters contains zinc. p. Some of the pennies are not Indian heads. 7
13. Symbolize the following sentences in PL using the given symbolization key. UD: By: Ry: Gy: Ly: Cy: Sy: Oy: The jellybeans in a larger glass jar y is black y is red y is green y is licorice-flavored y is cherry-flavored y is sweet y is sour a. All the black jellybeans are licorice-flavored. b. All the red jellybeans are sweet. c. None of the red jellybeans is licorice-flavored. d. Some red jellybeans are cherry-flavored. e. Some jellybeans are black and some are red. f. Some jellybeans are sour and some are not. g. Some jellybeans are black and some are red, but none is both. h. The red jellybeans are sweet, and the green jellybeans are sour. i. Some jellybeans are black, some are sweet, and some are licorice-flavored. j. No jellybeans are red and licorice-flavored. k. All the cherry-flavored jellybeans are red, but not all the red jellybeans are cherryflavored. 1. Every jellybean is red, and some are cherry-flavored and some are not cherryflavored. m. Every jellybean is red or every jellybean is black or every jellybean is green. n. Not all the jellybeans are licorice-flavored, but all those that are, are black. o. Some red jellybeans are sweet and some are not. p. Some jellybeans are sweet and some are sour, but none is sweet and sour. q. Some of the jellybeans are sour, but none of the licorice ones is. 8
14. Symbolize the following sentences in PL using the given symbolization key. UD: Persons Fx: x is a real estate agent Lx: x is a lawyer Px: x is a professor Nx: x lives next door lx: x is rich Sx: x can sell to yuppies Yx: x is a yuppie Rxy: x respects y f: Fred a. All real estate agents are yuppies. b. No real estate agents are yuppies. c. Some but not all real estate agents are yuppies. d. Some real estate agents are yuppies and some are not. e. If any real estate agent is a yuppie, all lawyers are. f. Any real estate agent who isn't a yuppie isn't rich. g. If any real estate agent can sell to yuppies, he or she is a yuppie. h. If any real estate agent can sell to yuppies, Fred can. i. Anyone who is a lawyer and a real estate agent is a yuppie and rich. j. Yuppies who aren't rich don't exist. k. Real estate agents and lawyers are rich if they are yuppies. 1. If Fred is a yuppie he's not a professor, and if he's a professor he's not rich. m. No professor who isn't rich is a yuppie. n. No professor who is self-respecting is a yuppie. o. Every self-respecting real estate agent is a yuppie. p. Real estale agents and lawyers who are rich are self-respecting. q. Real estate agents and lawyers who are either rich or yuppies are self-respecting. r. A yuppie who is either a real estate agent or a lawyer is self-respecting. s. A yuppie who is both a lawyer and a real estate agent is self-respecting. t A yuppie who is both a lawyer and a real estate agent lives next door. 9
15. Symbolize the following sentences in PL using the given symbolization key. UD: The employees of this college Exy: x earns more than y Dxy: x distrusts y Fx: x is a faculty member Ax: x is an administrator Cx: x is a coach Ux: x is a union member Rx: x should be ftred Mx: x is an MD Px: x is paranoid Ox: x is a union officer p: the president j: Jones a. Every administrator earns more than some faculty member, and every faculty member earns more than some administrator. b. If any administrator earns more than every faculty member, Jones does. c. No faculty member earns more than the president. d. Any administrator who earns more than every faculty member should be fired. e. No faculty member earns more than the president, but some coaches do. f. Not all faculty members are union members, but all union members are faculty members. g. No administrator is a union member, but some are faculty members. h. Every faculty member who is an administrator earns more than some faculty members who are not administrators. i. At least one administrator who is not a faculty member earns more than every faculty member who is an administrator. j. Every faculty member who is an MD earns more dian every faculty member who is not an MD. k. Some faculty members distrust every administrator, and some administrators distrust every faculty member. 1. There is an administrator who is a faculty member and distrusts all administrators who are not faculty members. m. Anyone who distrusts everyone is either paranoid or an administrator or a union officer. n. Everyone distrusts someone, but only administrators who are not faculty members distrust everyone. 10
16. Use the following symbolization key to translate sentences a-r into fluent English. (Note: All of the following claims are true.) UD: Positive integers Dxy: the sum of x and y is odd Exy: x times y is even Lxy: x is larger than y Oxy: x times y is odd Sxy: x plus y is even Ex: x is even Ox: x is odd Px: x is prime Pxy: x times y is prime a: 1 b: 2 c: 3 a. ( x)[ex ( y)exy] b. ( x)( y)[(ox & Oy) Oxy] c. ( x)( y)[sxy [(Ex & Ey) v (Ox & Oy)]] d. ( x)[(px & ( y)(py & Lxy)) Ox] e. ~ ( y)[py & ( x)(px Lyx)] f. ( y)( z)([(py & Pz) & (Lyb & Lzb)] Oyz) g. ~ ( x)( y)[(px & Py) & Pxy] h. ( x)(px & Ex) i. ( x)[px & ( y)eyx] j. ~ ( x)( y)lxy & ( x)( y)lyx k. ( x) ( y) [Oxy (Ox & Oy)] 1. ( x)( y)[exy (Ex v Ey)] m. ( x) ( y) [(Ox & Oy) (Oxy & Sxy)] n. ( x) ( y) (Lxy ~ Lyx) o. ( x)( y)[(ox & Ey) (Dxy & Exy)] p. ( x)( y)[[(px & Py) & Lcx] Exy] q. ( y)[(lya & Lcy) & (Py & Ey)] r. ( x)[(px & Ex) & ( y)((py & Lyx) Oy)] 11