Section 5.04 Self-Quiz

. Adriana is a better dancer than Rene. (a: Adriana; r: Rene; Dx: x is a dancer; Bxy: x is a better dancer than y)

Bra • (Da ∨ Dr)

Da • Bar

(Da ∨ Dr) • Bar

Da • Dr • Bar

Da ∨ Dr ∨ Bar

. Adriana is the best dancer. (a: Adriana; Dx: x is a dancer; Bxy: x is a better dancer than y)

Da • (∀x)[(Dx • x≠a) ⊃ Bax]

(∀x)(Dx ⊃ Bax)

~(∃x)(Dx • Bax)

~(∃x)(~Dx • Bxa)

Da • (∀x)(Dx • Bax)

. There are exactly two dancers better than Adriana. (a: Adriana; Dx: x is a dancer; Bxy: x is a better dancer than y)

(∃x)(∃y)(∀z){[Dz ⊃ (x=z ∨ y=z)] • Da • Bxa • Bya}

(∃x)(∃y)(∀z)[(x≠y • Bxa • Bya • Dx • Dy • Dz) ⊃ (z=x ∨ z=y)]

(∃x)(∃y){Dx • Bxa • Dy • Bya • x≠y • (∀z)[(Dz • Bza) ⊃ (z=x ∨ z=y)]}

~(∃x)(∃y)(∃z)(Bxa • Bya • Bza • x≠y • x≠z • y≠z • Dx • Dy • Dz)

(∃x)(∃y){x≠y • Dx • Dy • Bxa • Bya • (∀z){[Dz • (Bzx ∨ Bzy)] ⊃ z=a}}

. The discoverer of Polonium is Polish. (Px: x discovered Polonium; Qx: x is Polish)

(∃x)(Px • Qx)

(∃x)[Px • (∀y)(Py ⊃ y=x) • Qx]

(∀x)(Qx ⊃ Px)

(∀x)(Px ⊃ Qx)

(∃x)[Px • Qx • (∀y)(Qy ⊃ y=x)]

.

There are at least two newspapers in Jamal's Newsstand. (j: Jamal's Newsstand; Nx: x is a newspaper; Ixy: x is in y)

(∃x)(∃y)(Nx • Ny • Ixj • Iyj)

(∀x)(∀y)(∀z)[(Nx • Ny • Nz • Ixj • Iyj • Izj) ⊃ (x=y ∨ y=z ∨ x=z)]

(∃x)(∃y)(Nx • Ixj • Ny • Iyj • x≠y)

(∃x){Nx • Ixj • (∀y)[(Ny • Iyj) ⊃ x=y]}

(∃x)(∃y){Nx • Ixj • Ny • Iyj • x≠y • (∀z)[(Nz • Izy) ⊃ (x=z ∨ y=z)]}

.

There are exactly two newspapers in Jamal's Newsstand. (j: Jamal's Newsstand; Nx: x is a newspaper; Ixy: x is in y)

(∀x)(∀y)(∀z)[(Nx • Ny • Nz • x≠y • x≠z • y≠z) ⊃ (~Ixj ∨ ~Iyj ∨ ~Izj)]

~(∃x)(∃y)(∃z)(Nx • Ny • Nz • x≠y • x≠z • y≠z • Ixj • Iyj • Izj)

(∃x)(∃y){Nx • Ny • x≠y • (∀z)[(Nz ⊃ (z=x ∨ x=y)] • Ixj • Iyj}

(∃x)(∃y){Nx • Ixj • Ny • Iyj • x≠y • (∀z)[(Nz • Izj) ⊃ (z=x ∨ z=y)]}

(∃x)(∃y)(∀z){(Nz • z≠y • z≠x) ⊃ [(Ixj • Iyj • Nx • Ny) ⊃ ~Izj]}

.

There are exactly three newspapers in Jamal's Newsstand. (j: Jamal's Newsstand; Nx: x is a newspaper; Ixy: x is in y)

(∃x)(∃y)(∃z){Nx • Ixj • Ny • Iyj • Nz • Izj • x≠y • x≠z • y≠z •(∀w)[(Nw • Iwj) ⊃ (w=x ∨ w=y ∨ w=z)]}

(∃x)(∃y)(∃z){Nx • Ixj • Ny • Iyj • Nz • Izj • x=y • x=z • y=z •(∀w)[(Nw • Iwj) ⊃ (w≠x ∨ w≠y ∨ w≠z)]}

(∃x)(∃y)(∃z){Nx • Ny • Nz • Ixj • Iyj • Izj • x=y • x=z • y=z •(∀w)[(Nw • Iwj) ⊃ (w=x ∨ w=y ∨ w=z)]}

(∃x)(∃y)(∃z){Nx • Ny • Nz • Ixj • Iyj • Izj • x=y • x=z • y=z •(∀w)[(w=x ∨ w=y ∨ w=z) ⊃ (Nw • Iwj)]}

(∃x)(∃y)(∃z){Nx • Ny • Nz • Ixj • Iyj • Izj • x≠y • x≠z • y≠z •(∀w)[(w=x ∨ w=y ∨ w=z) ⊃ (Nw • Iwj)]}

.

The owner of Giovanni's Pizza is kind. (g: Giovanni's Pizza; Kx: x is kind; Oxy: x owns y)

(∃x)[Oxg • (∀y)(Oyg ⊃ y=x) • Kx]

(∀x)(Oxg ⊃ Kx)

(∀x)(Kx ⊃ Oxg)

(∀x)(∀y)[(Oxg • Oyg) ⊃ (x=y • Ky)]

~(∃x)(Oxg • ~Kx)

. Everyone except Katalin has been to New York. (k: Katalin; n: New York; Px: x is a person; Hxy: x has been to y)

~Hkn • (∀x)(Px ⊃ Hxn)

(∀x)[(Px • x≠k) ⊃ Hxn] • ~Hkn

~Hkn • (∀x)(Hxn ⊃ Px)

(∀x)[(Px • Hxn) ⊃ x≠k] • ~Hkn

~(∃x)(Px • x=k • Hxn) • ~Hkn

. Everyone except Katalin and Alice have been to New York. (a: Alice; k: Katalin; n: New York; Px: x is a person; Hxy: x has been to y)

~Hkn • ~Han • (∀x)[Px ⊃ (Hxn ∨ x≠a ∨ x≠k)]

~Hkn • (∀x)[(Px • x≠k • x≠a) ⊃ Hxn] • ~Han

(∀x)[Px ⊃ (Hxn ∨ x=a ∨ x=k)]

(∀x)[(x≠a • x≠k) ⊃ (Px • Hxn)]

(∀x)[(x≠a • x≠k) ⊃ (Px ⊃ Hxn)]

. There are at least two people shorter than Louisa. (l: Louisa; Px: x is a person; Sxy: x is shorter than y)

(∃x)(∃y)(x≠l • y≠l • Px • Py • Syl • Sxl)

(∃x)(∃y)(∀z)[z=l ⊃ (Syz • Sxz • x≠y)]

(∃x)(∃y)(Px • Py • Pl • Syl • Sxl)

(∃x)(∃y)[(x≠y • Px • Py) ⊃ (Sxl • Syl)]

(∃x)(∃y)(Px • Sxl • Py • Syl • x≠y)

. There are at least three people shorter than Louisa. (l: Louisa; Px: x is a person; Sxy: x is shorter than y)

(∃x)(∃y)(∃z)(x≠y • z=l • Px • Py • Pl • Sxz • Syz)

(∃x)(∃y)(∀z)[(Sxz • Syz • Px • Py) ⊃ z=l]

(∃x)(∃y)(∃z)(Px • Sxl • Py • Syl • Pz • Szl • x≠y • x≠z • y≠z)

(∃x)(∃y)(∃z)(Px • Py • x≠y • Pz • x≠z • Sxl • Syl • Szl • Pl)

(∃x)(∃y)(∃z)[Px • Py • Pz • Sxl • Syl • Szl • ~(x=y • y=z • x=z)]

. There is exactly one talented singer in Potsdam. (p: Potsdam; Sx: x is a singer; Tx: x is talented; Ixy: x is in y)

(∃x){Sx • Ixp • Tx • (∀y)[~y=x ⊃ ~(Sy ∨ Ty ∨ Iyp)]}

~(∃x)(∃y)(x≠y • Sx • Sy • Tx • Ty • Ixp • Iyp)

(∀x)(∃y)[(Sx • Px • Ixp) ⊃ (Sy • Py • Iyp • x=y)]

(∃x){Sx • Tx • Ixp • (∀y)[(Sy • Ty • Iyp) ⊃ y=x]}

(∃x){Sx • Ixp • Tx • (∀y)[(Ty • Sy) ⊃ (y≠x • ~Iyp]}

. The singer of Formation is famous. (f: Formation; Fx: x is famous; Sxy: x sings y)

(∃x)[Sxf • (∀y)(Syf ⊃ y=x) • Fx]

(∃x){Fx • (∀y)[Syf ⊃ (Sxf • y=x)]}

(∃x){Sxf ⊃ [Fx • (∀y)(Syf ⊃ y=x)]}

(∃x)[Sxf • Fx • (∀y)(x=y ⊃ Syf)]

(∃x)(∀y)[(Syf • Sxf) ⊃ (x=y • Fx)]

. Everyone except Lara watches The Voice. (l: Lara; a: The Voice; Px: x is a person; Wxy: x watches y)

(∀x)(Px • Wxa • x≠l)

(∀x)[(Px • Wxa) ⊃ x≠l]

~Wla • ~(∀x)(Px • Wxa)

âŒWla • (∀x)[(Px • x≠l) ⊃ Wxa]

~Wla • ~(∃x)(Px • ~Wxa)

. At least three squirrels eat blueberries. (Bx: x is a blueberry; Sx: x is a squirrel; Exy: x eats y)

(∃x)(∃y)(∃z)(∃w)(Sx • Sy • Sz • Bw • x≠y • x≠z • y≠z • Exw • Eyw • Ezw)

(∃x)(∃y)(∃z){Sx • Sy • Sz • x≠y • x≠z • y≠z • (∀w)[Bw ⊃ (Exw • Eyw • Ezw)]}

~(∃x)(∃y)(∃z)(∃t)(∃w)(Sx • Sy • Sz • St • x≠y • x≠z • x≠t • y≠z • y≠t • z≠t • Bw • Exw •Eyw • Ezw • Etw)

~(∃x)(∃y)(∃z)(∃t)[Sx • Sy • Sz • St • x≠y • x≠z • x≠t • y≠z • y≠t • z≠t • (∃w)(Bw • Exw) • (∃w)(Bw • Eyw) • (∃z)(Bz • Ezw) • (∃t)(Bt • Etw)]

(∃x)(∃y)(∃z)[Sx • (∃w)(Bw • Exw) • Sy • (∃w)(Bw • Eyw) • Sz • (∃w)(Bw • Ezw) • x≠y • x≠z • y≠z]

. There is a tree in which there are at most two squirrels. (Sx: x is a squirrel; Tx: x is a tree; Ixy: x is in y)

(∃x)[Tx • ~(∃y)(∃z)(∃t)(∃w)(Sy • Sz • St • Sw • Iyx • Izx • Itx • Iwx • y≠z • y≠t • y≠w • z≠t • z≠w • t≠w)]

(∃x){Tx • (∀y)(∀z)(∀w)[(Sy • Sz • Sw • Iyx • Izx • Iwx) ⊃ (y=z ∨ y=w ∨ z=w)]}

(∃x){Tx • (∀y)(∀z)(∀w)[(Sy • Iyx • Sz • Izx • Sw • Iwx) ⊃ (y=z • y=w • z=w)]}

(∀x)(∀y)(∀z){[Sx • (∃w)(Tw • Ixw) • Sy • (∃w)(Tw • Iyw) • Sz • (∃w)(Tw • Ixw)] ⊃ (x=y ∨ y=z ∨ x=z)}

(∃x)(∃y)(∃z)[(Sx • Sy • Sz • x≠y • y≠z • x≠z) ⊃ ~(∃w)(Tw • Ixw • Iyw • Izw)]

. There are exactly two professors who teach Logic. (l: Logic; Px: x is a professor; Txy: x teaches y)

~(∃x)(∃y)(∃z)(Px • Py • Pz • Txl • Tyl • Tzl)

(∃x)(∃y){Px • Py • x≠y • (∀z)[Pz ⊃ (x=y ∨ z=y)] • Txl • Tyl}

(∃x)(∃y){Px • Txl • Py • Tyl • x≠y • (∀z)[(Pz • Tzl) ⊃ (z=x ∨ z=y)]}

(∃x)(∃y){Txl • Px • Py • Tyl • (∀z)[(Pz • Tzl) ⊃ (z=x • z=y)] • x=y}

(∃x)(∃y){Px • Py • (∀z)(Tzl ⊃ x=z ∨ y=z)}

. The professor who has the biggest class is Sanda. (s: Sanda; Cx: x is a class; Px: x is a professor; Bxy: x is bigger than y; Hxy: x has y)

Ps • (∃x){Cx • Hsx • (∀y)(∀z)[(Py • Cz • Hyz • z≠x) ⊃ Bxz]}

(∀x){[Cx • (∃y)(Py • Hyx)] ⊃ (∃y)(Cy • Hsy • Byx)]

(∃x)(∃y){Px • Hxy • Cy • (∀z)(∀t)[(Pz • Hzt • Ct) ⊃ Byt] • x=s}

(∃x){Cx • Hsx • (∀y)[(Cy • x≠y) ⊃ Bxy]}

Ps • (∃x){Cx • Hsx • (∀y){y≠s ⊃ (∀z)[(Cz • Hyz) ⊃ Bxz]}}

. There are at most three psychology majors in Psych 210. (p: Psych 210; Px: x is a psychology major; Ixy: x is in y)

(∀w)(∀x)(∀y)(∀z)[(Pw • Px • Py • Pz • Iwp • Ixp • Iyp • Izp) ⊃ (w=x ∨ w=y ∨ w=z)]

~(∃w)(∃x)(∃y)(∃z)[(Pw • Px • Py • Pz • Iwp • Ixp • Iyp • Izp) ⊃ (w=x ∨ w=y ∨ w=z)]

~(∃w)(∃x)(∃y)(∃z)[Pw • Px • Py • Pz • Iwp • Ixp • Iyp • Izp • (w≠x ∨ w≠y ∨ w≠z ∨ x≠y ∨ x≠z ∨ y≠z)]

(∀w)(∀x)(∀y)(∀z)[(Pw • Iwp • Px • Ixp • Py • Iyp • Pz • Izp) ⊃ (w≠x ∨ w≠y ∨ w≠z ∨ x≠y ∨ x≠z ∨ y≠z)]

(∀w)(∀x)(∀y)(∀z)[(Pw • Iwp • Px • Ixp • Py • Iyp • Pz • Izp) ⊃ (w=x ∨ w=y ∨ w=z ∨ x=y ∨ x=z ∨ y=z)]

Quiz Content