# Equivalence Relations: Solving & Proving Reflexivity, Symmetry & Transitivity

• MHB
• Lancelot1
In summary, an equivalence relation is a mathematical concept that defines a relationship between two objects or elements. It has three properties: reflexivity, symmetry, and transitivity. Reflexivity states that every element is related to itself, symmetry states that if two elements are related, then they are also related in reverse, and transitivity states that if two elements are related and one of them is related to a third element, then the first element is also related to the third element. To prove an equivalence relation, all three properties must be satisfied.
Lancelot1
Dear All,

I am trying to solve the attached two questions.

In both I need to determine if the relation is an equivalence relation, to prove it if so, and to find the equivalence classes.

In both cases it is an equivalence relation, and I managed to prove both relations are reflexive. Now I am with symmetry and transitivity. In the first case, it is obviously symmetric, but how do I prove it ? Same for the second relation, I find it hard to prove and also to prove transitivity.

For reflexivity, m-m = 0 which is dividable by 4. Also 7m-5m=2m, which is dividable by 2 and therefore even.

Thank you !

Lancelot said:
Dear All,

I am trying to solve the attached two questions.

In both I need to determine if the relation is an equivalence relation, to prove it if so, and to find the equivalence classes.

In both cases it is an equivalence relation, and I managed to prove both relations are reflexive. Now I am with symmetry and transitivity. In the first case, it is obviously symmetric, but how do I prove it ? Same for the second relation, I find it hard to prove and also to prove transitivity.

View attachment 11885

For reflexivity, m-m = 0 which is dividable by 4. Also 7m-5m=2m, which is dividable by 2 and therefore even.

Thank you !
Some pointers.

For the first problem. Look at it this way. If mEn then 4|m - n. That means m - n = 4k, where k is some integer. If this is true then can you find such a k' for n - m? Any integer k' will do. Transitivity works on the same model.

For the second we have a similar idea. If mRn then 2|7m - 5n. So 7m - 5n = 2k. What can we say about the parities of m and n here? (ie. are they odd/even?) When this is true what can we say about 7n - 5m? Transitivity here isn't too bad. If we have mRn and nRp then we know that 7m - 5n = 2j and 7n - 5p = 2k. So again, what is the parity of m and n? Of n and p? So then what are the parities of m and p? So then what can we say about 7m - 5p?

Give it a go and if you are still having problems come back and show us what you've been able to do with it.

-Dan

Yes, you have shown that both are 'reflexive', mEm and mRm.

An "equivalence relation" must also be symmetric. We need to show that "if mEn then nEm" and "if mRn the nRm".
If mEn then 4 divides m- n so m- n= 4k for some integer k. Now n- m= -(m- n)= -4k= 4j where j=-k.
If mRn 7m- 5n is even so 7m- 5n= 2k for some integer k. 7m= 2k- 5n so that 7n- 5m= (7m- 5n)- (12m- 2n)= 2k- 2(6m- n)= 2(k- 6m+ n) an even number.

An "equivalence relation" must also be "transitive"- if mEn and nEp then mEp and if mRn and nRp then mRp.
If mEn then 4 divides m- n so m- n= 4k for some integer k.
If nEp then 4 divides n- p so n- p= 4k' for some integer k'.
m- p= (m- n)- (n- p)= 4k- 4k'= 4(k- k) so 4 divides m- p.

If mRn then 7m- 5n= 2k for some integer k.
If nRp then 7n- 5p= 2k' for some integer k'.
7m- 5p= (7m- 5n)+ (7n- 5p)- 2n= 2k+ 2k'- 2n= 2(k+ k'- 1) so is even.

## 1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between two elements in a set. It is a relation that is reflexive, symmetric, and transitive.

## 2. How do you prove reflexivity in an equivalence relation?

To prove reflexivity in an equivalence relation, you must show that every element in the set is related to itself. This can be done by showing that for any element a in the set, (a,a) is an element of the relation.

## 3. What is the importance of symmetry in an equivalence relation?

Symmetry is important in an equivalence relation because it ensures that the relationship is bi-directional. This means that if element a is related to element b, then element b is also related to element a.

## 4. How can you determine transitivity in an equivalence relation?

To determine transitivity in an equivalence relation, you must show that if element a is related to element b, and element b is related to element c, then element a is also related to element c. This can be done by showing that if (a,b) and (b,c) are elements of the relation, then (a,c) is also an element of the relation.

## 5. Can you give an example of an equivalence relation?

One example of an equivalence relation is the relation "is the same age as" on a group of people. This relation is reflexive, symmetric, and transitive. For example, person A is the same age as person A (reflexive), if person A is the same age as person B, then person B is also the same age as person A (symmetric), and if person A is the same age as person B, and person B is the same age as person C, then person A is also the same age as person C (transitive).

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