Assume that R is a relation on the set Z of integers and it is given by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$. Then, R is

The relation R is defined by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$

Two integers x and y are related if the distance between them is at most 1. This means x and y can be the same number (distance = 0) or adjacent numbers (distance = 1).

For example: (5, 5), (5, 6), (7, 8), (10, 9) are all in R, but (5, 7) is not in R because $|5 - 7| = 2 > 1$

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For R to be reflexive, $(x, x) \in R$ must hold for any integer x.

$|x - x| = |0| = 0 \leq 1$

R is reflexive.

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For R to be symmetric, if $(x, y) \in R$, then $(y, x) \in R$ must also hold.

If $(x, y) \in R$, then $|x - y| \leq 1$

$|y - x| = |x - y| \leq 1$

R is symmetric.

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For R to be transitive, if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$ must hold.

Consider x = 1, y = 2, z = 3:

$|1 - 2| = 1 \leq 1$, so $(1, 2) \in R$

$|2 - 3| = 1 \leq 1$, so $(2, 3) \in R$

$|1 - 3| = 2 \not\leq 1$, so $(1, 3) \notin R$

Even though 1 relates to 2 and 2 relates to 3, 1 does not relate to 3.

R is not transitive.

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R is reflexive and symmetric but not transitive.