Question
Asked Oct 18, 2019
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1. Consider the following rules on the set of natural numbers, N, 1) write down 5
ordered pairs of natural numbers that are related under the relation, and 2) decide for
the properties of reflexiveness (and irreflexiveness), symmetry (and antisymmetry) and
transitivity. Note: please specify explicitly whether the relation has each of the five
properties or not, and explain your answer.
"is less than or equal to"
a factor of"
c) "is 3 less than"
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1. Consider the following rules on the set of natural numbers, N, 1) write down 5 ordered pairs of natural numbers that are related under the relation, and 2) decide for the properties of reflexiveness (and irreflexiveness), symmetry (and antisymmetry) and transitivity. Note: please specify explicitly whether the relation has each of the five properties or not, and explain your answer. "is less than or equal to" a factor of" c) "is 3 less than"

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Expert Answer

Step 1

The 5 order pair of the natural number that is satisfy the relation “less than or equal to” as follows,

That means, if a and b are the natural number then it satisfy (a b)

So, (1, 1), (1, 2), (2, 2), (2, 3), (1, 3)

When a, b and c belongs to the natural number or real number then,

If (a, a) or (b, b) or (c, c) ϵ N, then it reflexive.

If (a, b) ϵ N and also (b, a) ϵ N then it symmetry.

And, (a, b) ϵ N and (b, c) ϵ N then if (a, c) ϵ N, it is transitive.

Now, according to the above properties,

Since (1, 1) and (2, 2) are the reflexive. So the given relation is reflexive.

The above relation are not symmetric because a less than or equal to b but b not less than or equal to a.

Here, (1, 2) and (2, 3) and then also (1, 3) so the above relation is transitive.

Step 2

The 5 order pair of the natural number that is satisfy the relation “is a factor of” as follows,

That means, if a and b are the natural number then it satisfy (a factor of b)

So, (2, 4), (2, 6), (...

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