9.1ex I
Symbolizing with Identity
Multiple Choice: Pick the correct symbolization for each problem below. Use the following interpreation:

UD: coins in my pocket
Px: x is a penny
Qx: x is a quarter
Sx: x is silver
m: my half dollar

1. There is at least one quarter (in my pocket).
a. (%x)Qx b. (%x)Qx&(%x)~Qx c. (^x)(Qxv~Qx)

2. There is at least one quarter but no penny.
a. (%x)Qx b. (%x)Qx&~(%x)Px c. (^x)(Qxv~Px)

3. There are at least two pennies.
a. (%x)Px&(%y)Py&~Ixy b. (%x)Qx&~(%x)Px c. (^x)(Px>(Py&~Ixy)) d. (%x)(%y)(~Ixy&(Px&Py))

4. There are exactly two quarters.
a. (%x)(%y)(^z)[Qz=(IzxvIzy)] b. (%x)(%y)(^z)[~Ixy&(Qz=(IzxvIzy))] c. (%x)(%y)[~Ixy&(Qx&Qy)]

5. There are exactly two quarters. (For the second time!)
a. ~(%x)(%y)(^z)[Qz=(IzxvIzy)] b. (^x)(^y)(^z)[(Qx&Qy)&Qz)>((IxyvIxz)vIyz)] c. (%x)(%y)[~Ixy&(Qx&Qy)]&(^x)(^y)(^z)[(Qx&Qy)&Qz)>((IxyvIxz)vIyz)]

6. The quarter in my pocket is made of silver.
a. (%x)(Qx&Sx) b. ~(%x)(%y)(~Ixy&(Qx&Qy)) c. (%x)[(Qx&(^y)(Qy>Iyx)&Sx]

7. All coins in my pocket except my half-dollar are pennies.
a. (^x)(Cx>Ixm) b. (^x)(~Ixm>Px) c. (^x)(Ixm=Px)

9.1ex II
Symbolization with Identity

Symbolize the following.

UD: all people
b: Bill Clinton
g: George W. Bush
u: the United States
s: the Supreme Court
Cx: x is a country
Fx: x is female
Pxy: x is president of y
Ixy: x = y
Lxy: x leads y

  1. George is US president and Bill is not. (Hint)
  2. George is the (one and only) president of the Supreme Court. (Hint)
  3. There is more than one president of the US. (Hint)
  4. There is exactly one president of the US. (Hint)
  5. There is no president of the US.
  6. There are at least two countries.(Hint)
  7. There is at most one country. (Hint)
  8. At least one female leads a country. (Hint)
  9. At most one person leads the US. (Hint)
  10. No one but George leads the US. (Hint)

9.1ex III
Symbolization with Identity

Symbolize the following.

UD: everything in the White House.
g: George W. Bush
Dx: x is a dog
Lx: x is large
Px: x is a person
Rx: x is a Rottweiler
Oxy: x owns y
Ixy: x = y

  1. George owns exactly one dog (living in the White House).
  2. Someone owns at least one Rottweiler.
  3. George owns no Rottweilers.
  4. There is exactly one dog owned by George.
  5. There are at least two dogs living in the White House.
  6. Everything living in the White House is a person except for one Rottweiler.
  7. There is at most one dog living in the White House.
  8. George's dog is a Rottweiler. (Hint)
  9. The Rottweiler owned by George is large.
  10. All people except George are dog owners.
Q9.2
Derivations in PDI

1. From the set of premises:

      (Iab&Iba)&Ibc

   derive the conclusion:

      Iac

2. From the set of premises:

      (^x)(Gcd>Txd)
      Ikc&Gkd

   derive the conclusion:

      Tad

3. Show that

      (^x)Ixx

   is logically true by giving a derivation.

4. Show that

      Iba

   is logically equivalent to

      Iab

   by providing two appropriate derivations.

      

9.2ex I
Derivations in PDI

1. From the set of premises:

      Iab&Sbc
      (^x)(Sac>Z)

   derive the conclusion:

      Z

2. Show that

      (^x)(^y)(Ixy=Iyx)

   is logically true by giving a derivation.

3. Show that

      (%x)~Ixx

   is logically false by giving a derivation.

4. Show that

      Iab&Iba

   is logically equivalent to

      Iab

   by providing two appropriate derivations.

5. From the set of premises:

      (^x)(^y)(Gxy>~Ixy)
      Iab

   derive the conclusion:

      ~Gba

 

9.3ex I
PLIF: Predicate Logic with Identity and Functions
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. Which of the following terms is well formed?
a. $x b. *x c. *xx d. $$x

2. Which of the following terms is well formed?
a. **a b. *aa c. aaa d. ***

3. Which of the following is a formula of PLIF?
a. **a b. **a>**z c. Q**a>P**z d. **a>P**z

4. Which of the following is a formula of PLIF?
a. A>B b. A{xyz}>B{xyz} c. A{x$y$z}|Bxyz

5. Which of the following is NOT a formula of PLIF?
a. Gxy=T{z+k}*a b. Gxy=T{z+k}a
c. Gxy=T*{z+k}*a d. Gxy=*T{z+k}*a

6. Which of the following is a SENTENCE of PLIF
a. (^x)(Tx>(%y)G{x|y}az) b. (^x)(^z)(Txy>(%y)G{x|y}az)
c. (^x)(^z)(Txz>(%y)G{x|y}az) d. None of the above.

9.4ex I
Group Theory Proofs

Derivations are a little too difficult to do on the computer at this stage. So, try to write out proof to the following the old fashioned way: on paper!

Some of these require that you follow the dictates of PLIF and PDI. Others you may want to do less formally. Use the tutorials examples to give you an idea of how to proceed. Then try to have fun!

  1. Theorem 1 says that if e and f are identity elements, then e=f. But before we pick 'e' as our name of the one and only identity element, we need to prove that there is exactly one identity element. The proof is easy from theorem 1 and one of the axioms of group theory. But you should state it informally. (Feel free to take theorem 1 as a premise of your proof.)
  2. Now, turn your informal proof of problem 1 into a PDI proof.
  3. Tutorial four ends with the theorem that there can be only one inverse for an element of a group. That's the informal statement. Give the statement of this theorem more formally in our language PLIF.
  4. It's a bit long, but try to give the derivation which proves your statement from 1 using the rules of PDI in using the language PLIF.
  5. Show that interpretation 4 really is a group. Explain why these sorts of motions satisfy the three axioms of group theory.
  6. Explain in your own words why Interpretation 2 is not a group.
  7. All the groups we studied in the tutorial satisfy this additional axiom:
        (^x)(^y)I{x+y}{y+x}
    All such groups are called "abelian". Now consider a new interpretation: Interpretation 5. This interpretation is just like the motions of Interpretation 4 except that it includes projections onto a plane. A projection onto a plane just takes a two dimensional object in 3 space and transforms each point of the object not on the plane to the point on the plane closest to the original. Show that the collection of motions generated by translations, rotations and projections is not abelian.