Let's analyze the relation $B$ on the set $A = \{1, 2, 3, 4, 5\}$ to answer each of the six questions:

The relation $B$ is defined as: $B = \{(1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 1), (5, 3), (5, 4), (5, 5)\}$

(i) Is the relation reflexive?

A relation is reflexive if for every element $x \in A$, $(x, x)$ is in the relation $B$.

For this set $A = \{1, 2, 3, 4, 5\}$, we need to check if $(1, 1)$, $(2, 2)$, $(3, 3)$, $(4, 4)$, and $(5, 5)$ are present in $B$.

• $(1, 1)$ is missing.
• $(2, 2)$ is present.
• $(3, 3)$ is present.
• $(4, 4)$ is present.
• $(5, 5)$ is present.

Since $(1, 1)$ is missing, the relation is not reflexive.

(ii) Is the relation symmetric?

A relation is symmetric if whenever $(x, y) \in B$, $(y, x)$ is also in $B$.

Checking pairs:

• $(1, 2)$ is in $B$, but $(2, 1)$ is also in $B$.
• $(1, 3)$ is in $B$, but $(3, 1)$ is also in $B$.
• $(1, 4)$ is in $B$, and $(4, 1)$ is in $B$.
• $(1, 5)$ and $(5, 1)$ are both in $B$.

Similarly, checking other pairs such as $(2, 3)$, $(3, 2)$, etc., we find that for every $(x, y)$, $(y, x)$ is present in the relation.

Thus, the relation is symmetric.

(iii) Is the relation anti-symmetric?

A relation is anti-symmetric if whenever $(x, y) \in B$ and $(y, x) \in B$, then $x = y$.

We already found symmetric pairs such as $(1, 2)$ and $(2, 1)$, $(1, 3)$ and $(3, 1)$, etc., where $x \neq y$, meaning that the relation is not anti-symmetric.

Therefore, the relation is not anti-symmetric.

(iv) Is the relation transitive?

A relation is transitive if whenever $(x, y) \in B$ and $(y, z) \in B$, then $(x, z) \in B$.

For example, take $(1, 2)$ and $(2, 3)$, we need to check if $(1, 3)$ is in $B$, which it is. We can similarly check other combinations, but there are counterexamples where this property does not hold for all combinations.

Thus, the relation is not transitive.

(v) Is the relation an equivalence relation?

A relation is an equivalence relation if it is reflexive, symmetric, and transitive.

Since we found that the relation is not reflexive and not transitive, the relation is not an equivalence relation.

(vi) Is the relation a partial order?

A relation is a partial order if it is reflexive, anti-symmetric, and transitive.

Since the relation is neither reflexive nor anti-symmetric, the relation is not a partial order.

Would you like further details or clarifications on any point?

Related Questions:

1. What are the conditions for a relation to be reflexive?
2. Can a relation be symmetric but not anti-symmetric?
3. How can we check for transitivity in a relation efficiently?
4. Can a relation be both an equivalence relation and a partial order?
5. What are real-world examples of symmetric and anti-symmetric relations?

Tip:

When checking for symmetry and anti-symmetry, always consider whether reversing the pair results in the same or a different outcome.