Need help please with 1 and 2 each relation in the following

Need help please with 1 and 2

each relation in the following set R: l(x. y) e N x NIx has the same number of prime factors as yh R, i(r,y) e N x N w does not have more prime factors than yh organize your answers to the following queri in a table with four columns with one row for i) Determine which of the following properties apply reflexive, symmetric, transitive. i) Decide whether the relation is an equivalence relatio 2) Define a relation, S, on z x Z by (a, b) e if and only if a b or a -b i) Show that S is an equivalence relation. ii) Describe the equivalence classes

Solution

The question 1 contains a lot of sub-parts to check 3 relations for reflexive, symmetric and transitive, i am solving the first problem in detail, post one more question to get the answer to second problem

Q1) For R1

The relation is said to be reflexive if (a,a) belongs to R

Since |x| = |x|, hence the relation is reflextive

The relation is symmetric if (a,b) belongs to R, then it implies (b,a) will also belong to R

since |x|=|y|, hence |y|=|x|

Therefore the relation is symmetric in nature

The releation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) belongs to R

since |x|=|y| and |y|=|z|, it implies |x| = |z|

Hence the relation is transitive in nature

The R1 relation is an equivalence relation since it is reflexive, symmetric and transitive in nature

For R2

The relation is said to be reflexive if (a,a) belongs to R

Since x has the same number of prime factors as x, hence the relation is reflexive

The relation is symmetric if (a,b) belongs to R, then it implies (b,a) will also belong to R

since x has same prime factor as y, then y will also have same prime factor as x

Therefore the relation is symmetric in nature

The releation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) belongs to R

since x and y have same prime factors, y and z have same prime factors, hence x and z will also have same prime factors

Hence the relation is transitive in nature

The R2 relation is an equivalence relation since it is reflexive, symmetric and transitive in nature

For R3

The relation is said to be reflexive if (a,a) belongs to R

Since x has the same number of prime factors as x, hence the relation is reflexive

The relation is symmetric if (a,b) belongs to R, then it implies (b,a) will also belong to R

since x doesnt have more prime numbers than y, hence the relation cannot be symmetric

Therefore the relation is not symmetric in nature

The releation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) belongs to R

Hence the relation is not transitive in nature

The R3 relation is not an equivalence relation since it is reflexive, not symmetric and neither transitive in nature

	Reflexive	Symmetric	Transitive	Equivalence
R1	Yes	Yes	Yes	Yes
R2	Yes	Yes	Yes	Yes
R3	Yes	No	No	No