Let's go through each problem in the provided image step-by-step.

Problem 6

Determine if each relation $R$ on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive. Relations are defined for pairs $(x, y)$ as follows:

Solution Analysis

1. Reflexive: $(x, x) \in R$ for all $x$.
2. Symmetric: If $(x, y) \in R$, then $(y, x) \in R$.
3. Antisymmetric: If $(x, y) \in R$ and $(y, x) \in R$, then $x = y$.
4. Transitive: If $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$.

For each condition (a-h), evaluate the above properties. For instance:

• a) $x + y = 0$:
  • Reflexive? No, as $x + x = 0 \Rightarrow x = 0$ (only zero is reflexive).
  • Symmetric? Yes, since if $x + y = 0$, then $y + x = 0$.
  • Antisymmetric? No, as $(x, y)$ and $(y, x)$ could both satisfy $x + y = 0$ without $x = y$.
  • Transitive? Yes, because if $x + y = 0$ and $y + z = 0$, then $x + z = 0$.

Repeat similarly for each condition.

Problem 7

Analyze each relation $R$ on the set of all integers for reflexive, symmetric, antisymmetric, and/or transitive. Relations are defined for pairs $(x, y)$ as follows:

For each condition (a-h), apply the properties:

• a) $x \neq y$:
  • Reflexive? No, as $x = x$ does not satisfy $x \neq y$.
  • Symmetric? Yes, if $x \neq y$ then $y \neq x$.
  • Antisymmetric? Yes, since $x = y$ would contradict the relation.
  • Transitive? No, as there’s no guarantee of transitivity.

Repeat for conditions $b$ through $h$.

Problem 8

Show that the relation $R = \emptyset$ on a nonempty set $S$ is symmetric and transitive but not reflexive.

• Reflexive? No, since $R$ is empty, there’s no $(x, x) \in R$ for any $x$.
• Symmetric? Yes, because there are no counterexamples in an empty relation.
• Transitive? Yes, again because there are no counterexamples.

Problem 9

Show that the relation $R = \emptyset$ on the empty set $S = \emptyset$ is reflexive, symmetric, and transitive.

On an empty set, the conditions for reflexivity, symmetry, and transitivity are vacuously satisfied because there are no elements to contradict any of these properties.

Problem 10

Give an example of a relation on a set that is:

• a) Both symmetric and antisymmetric.
• b) Neither symmetric nor antisymmetric.

Examples:

• a) The relation $R = \{(x, x)\}$ on a set $S$ (like all pairs where $x = y$) is both symmetric and antisymmetric.
• b) The relation $R = \{(1, 2), (2, 1)\}$ on a set $S = \{1, 2\}$ is neither symmetric nor antisymmetric.

Would you like a detailed solution for any specific part?

Follow-up Questions

1. How does the transitivity property change if the relation is defined differently?
2. Can you think of other relations that are only symmetric or only antisymmetric?
3. Why is the empty relation both symmetric and transitive?
4. How would these properties change if we used a finite set rather than the set of all real numbers?
5. What would happen if we added an additional condition to each relation?

Tip

Always verify each property (reflexive, symmetric, antisymmetric, transitive) independently to ensure clarity in understanding each relation's behavior.