Please answer all five question. I need this is the next 1 hour please. Leave sh

Question

Please answer all five question. I need this is the next 1 hour please. Leave short explanation for each answer, Thanks

Question 1

What are the properties of the following relation ?

Check all that apply.

Question 1 options:

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above.

Question 2

What are the properties of the following relation

{ <a,b>, <b,a>, <c,c>}

Check all that apply.

Question 2 options:

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above.

Question 3

What are the proporties of the following relation

{ <a,b>, <b,b>, <a,c>, <c,b>}

Check all that apply.

Question 3 options:

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above.

Question 4 (4 points)

What are the proporties of the following relation

{ <a,b>, <b,c>, <c,d>, <d,e>}

Check all that apply.

Question 4 options:

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above

Question 5

What are the proporties of the following relation

{ <a,b>, <b,c>, <c,d>, <a,c>, <a,a>,<b,b> }

Check all that apply.

Question 5 options:

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above

Reflexive

Transitive

Symmetric

Antisymmetric

None of the above.

Explanation / Answer

1)

Whether the empty relation is reflexive or not depends on the set on which you are defining this relation -- you can define the empty relation on any set XX.

The statement "R is reflexive" says: for each xX we have (x,x)R. This is vacuously true if X= and it is false if X is nonempty.

The statement "R is symmetric" says: if (x,y)R then (y,x)R. This is vacuously true, since (x,y)R(x,y)R for all x,yXx,yX.

The statement "R is transitive" says: if (x,y)R

and (y,z)R

then (x,z)R

. Similarly to the above, this is vacuously true.

To summarize, R is an equivalence relation if and only if it is defined on the empty set. It fails to be reflexive if it is defined on a nonempty set.

2)

Symmetric because for every (a, b), we have (b, a) (option 3)

3)

Antisymmetric because

for (a,b) there is no (b,a)

for (a,c) there is no (c,a)

for (c,b) there is no (b,c)

4)

Transitive

Transitivity requires that if (a,b) and (b,c) are present in the relation, then so is (a,c),

Here it is there

5)

Anyisymmetric

for (a,b) there is no (b,a)

for (a,c) there is no (c,a)

for (c,b) there is no (b,c)

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