Exercises for first-order logic

Determine whether the relation $R$ on the set of all people is reflexive,antireflexive, symmetric,antisymmetric, and/or transitive, where $\left(a,b\right)\in R$ if and only if …

1. $a$ is older than $b$ .

2. $a$ is at least as old as $b$ .

3. $a$ and $b$ are exactly the same age.

4. $a$ and $b$ have a common grandparent.

5. $a$ and $b$ have a common grandchild.

For each of the following, if the statement is true, explain why, and if the statement is false, give a counter-example relation.

1. If $R$ is reflexive, then $R$ is symmetric.

2. If $R$ is reflexive, then $R$ is antisymmetric.

3. If $R$ is reflexive, then $R$ is not symmetric.

4. If $R$ is reflexive, then $R$ is not antisymmetric.

5. If $R$ is symmetric, then $R$ is reflexive.

6. If $R$ is symmetric, then $R$ is antireflexive.

7. If $R$ is symmetric, then $R$ is not antireflexive.

Let $P(x)$ be the statement

has been to Prague

1. Express each of these quantifications in English.

   • $\exists x\colon P(x)$
   • $\forall x\colon P(x)$
   • $\neg \exists x\colon P(x)$
   • $\neg \forall x\colon P(x)$
   • $\exists x\colon \neg P(x)$
   • $\forall x\colon \neg P(x)$
   • $\neg \exists x\colon \neg P(x)$
   • $\neg \forall x\colon \neg P(x)$

2. Which of these mean the same thing?

Let $C(x)$ be the statement

$x$ has a cat

$x$ has a dog

$x$ has a ferret

1. A classmate has a cat, a dog, and a ferret.

2. All your classmates have a cat, a dog, or a ferret.

3. At least one of your classmates has a cat and a ferret, but not a dog.

4. None of your classmates has a cat, a dog, and a ferret.

5. For each of the three animals, there is a classmate of yours that has one.

Determine the truth value of each of these statements ifthe domain is all real numbers. Where appropriate, give a witness.

1. $\exists x\colon x^2=2$
2. $\exists x\colon x^2=-1$
3. $\forall x\colon x^2+2\ge 1$
4. $\forall x\colon x^2\neq x$

Let $P(x)$ , $Q(x)$ , $R(x)$ , and $S(x)$ be the statements

$x$ is a duck

$x$ is one of my poultry

$x$ is an officer

$x$ is willing to waltz

1. No ducks are willing to waltz.

2. No officers ever decline to waltz.

3. All my poultry are ducks.

4. My poultry are not officers.

5. Does the fourth item follow from the first three taken together? Argue informally; you don't need to use thealgebra or inference rules for first-order logic here.

You come home one evening to find your roommate exuberant becausethey have managed to prove that there is an even prime number bigger than two.More precisely, they have a correct proof of $\exists y\colon (P(y)\land (y> 2))\implies E(y)$ , for the domain of natural numbers,with $P$ interpreted as

is prime?

is even?

1. …and realize the formula is indeed true for that interpretation.Briefly explain why. You don't need to give a formal proof using Boolean algebra orinference rules; just give a particular value for $y$ and explain why it satisfies the body of

   $\exists y$

   .

2. Is the formula still true when restricted to the domain of natural numbers two or less?Briefly explain why or why not.

3. Is the formula still true when restricted to the empty domain? Briefly explain why or why not.

4. Give a formula that correctly captures the notion

   there is an even prime number bigger than 2

   .