Consider relation R3 on the set of positive real numbers where (x, y) is an elem

Question

Consider relation R3 on the set of positive real numbers where (x, y) is an element of R3 if and only if x/y is an element of Q. Decide whether it is reflexive, anti-reflexive, symmetric, anti-symmetric an transitive and show that this an equivalence relation. Describe the equivalence classes. What is the equivalence class of 2? What is the equivalence class of pi? Justify your answers.

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Consider relation R3 on the set of positive real numbers where (x, y) is an element of R3 if and only if x/y is an element of Q. Decide whether it is reflexive, anti - reflexive, symmetric, anti - symmetric an transitive and show that this an equivalence relation. Describe the equivalence classes. What is the equivalence class of 2? What is the equivalence class of pi? Justify your answers. ***Below is a screenshot of the problem for your convenience. Please answer completely.***

Explanation / Answer

Proof of reflexivity: Fix any positive real x. Then x/x is defined and is equal to 1. Since 1 is a rational number, (x,x) is in R3.

Disproof of antireflexivity: 1 is a positive real number and (1,1) is in R3.

Disproof of antisymmetry: (1,2) and (2,1) are in R3 but 1 is not equal to 2.

Proof of symmetry: Pick any positive reals x and y such that (x,y) in R3. Then x/y = n/m for some non-zero integers n and m. Since n and m are not zeros, y/x is defined and is equal to m/n. Since y/x can be represented as a ratio of two integers, it is a rational number. Therefore, (y,x) is in R3.

Proof of transitivity: Pick any positive reals x,y, and z such that (x,y) and (y,z) are in R3. Then x/y = m/n and y/z = p/q, where n,m,p,q are non-zero integers. Then x/z = (x/y)*(y/z) = (mp)/(nq). Since x/z can be represented as a ratio of two integers, its is a rational number. Therefore, (x,z) is in R3.

Proof of being equivalence relation: Since R3 is reflexive, symmetric, and transitive, it is an equivalence relation.

Equivalence classes: Equivalence class of a is the set of all positive reals x such that a/x is rational. This means that x is in the equivalence class of a if and only if x is the product of a and some positive rational number. In set-builder notation: {(m/n)*a | n,m are positive integers}.

Equivalence class of 2: is {(m/n)*2 | m and n are positive integers} = {m/n| n,m are positive integers}. In other words the equivalence class of 2 is the set of all positive rational numbers.

Equivalence class of Pi: is {(m/n)*Pi | m and n are positive integers}

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