Relations equations i found online and i'm tring to get the answers to understand

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November 22nd 2012, 07:21 PM
math333

Relations equations i found online and i'm tring to get the answers to understand

Determine if the following relations are reexive, symmetricand transitive.
If a property does not hold, say why.

let R= {(a,a) , (b,b) , (c,c) , (d,d) , (a,b) , (b,a)} be a relation on the setA={a,b,c,d}

let R= {(a,a) , (a,c) , (c,c) , (b,b) , (c,b) , (b,c)} be a relation on the setA={a,b,c}

let R= {(a,a) , (a,c) , (c,c) , (c,b) , (b,c)} be a relation on the setA={a,b,c}

let R= {(a,a) , (b,b) , (c,c) , (d,d)} be a relation on the set A={a,b,c,d}
2. Suppose R is a symmetric and transitive relation on a setA, and there is an element a ∈ A for which (a; x)∈Rfor all x∈ A. Prove that R is reflexive.

3. Prove or disprove: If a relation is symmetric and transitive, then it isalso reflexive.

4. Consider the relation R = {(x; y) : 3x - 5y is even} on ℤ.Prove R is an equivalence relation.

5. Consider the relation R = {(x; y) : x - 3y is divisible by 4} on ℤ.Prove R is an equivalence
relation.
November 22nd 2012, 07:24 PM
topsquark

Re: Relations equations i found online and i'm tring to get the answers to understand

Quote:

Originally Posted by math333

Determine if the following relations are reexive, symmetricand transitive.
If a property does not hold, say why.

let R= {(a,a) , (b,b) , (c,c) , (d,d) , (a,b) , (b,a)} be a relation on the setA={a,b,c,d}

let R= {(a,a) , (a,c) , (c,c) , (b,b) , (c,b) , (b,c)} be a relation on the setA={a,b,c}

let R= {(a,a) , (a,c) , (c,c) , (c,b) , (b,c)} be a relation on the setA={a,b,c}

let R= {(a,a) , (b,b) , (c,c) , (d,d)} be a relation on the set A={a,b,c,d}
2. Suppose R is a symmetric and transitive relation on a setA, and there is an element a ∈ A for which (a; x)∈Rfor all x∈ A. Prove that R is reflexive.

3. Prove or disprove: If a relation is symmetric and transitive, then it isalso reflexive.

4. Consider the relation R = {(x; y) : 3x - 5y is even} on ℤ.Prove R is an equivalence relation.

5. Consider the relation R = {(x; y) : x - 3y is divisible by 4} on ℤ.Prove R is an equivalence
relation.

Reflexive means xRx for all x.
Symmetric means xRy implies yRx.
Transitive means if xRy and yRz then xRz

What can you say about the different R's you are given?

-Dan
November 22nd 2012, 07:32 PM
math333

Re: Relations equations i found online and i'm tring to get the answers to understand

I think the first two would be symmetric the third one is transitive and the last is reflexive but how would you show your working for something like this thanks
November 23rd 2012, 06:24 AM
Plato

Re: Relations equations i found online and i'm tring to get the answers to understand

let $R= \{(a,a) , (b,b) , (c,c) , (d,d) , (a,b) , (b,a)\}$ be a relation on the set $A=\{a,b,c,d\}$
reflexive, symmetric, & transitive.

let $R= \{(a,a) , (a,c) , (c,c) , (b,b) , (c,b) , (b,c)\}$ be a relation on the set $A=\{a,b,c\}$
reflexive, not symmetric, & not transitive.

let $R= \{(a,a) , (a,c) , (c,c) , (c,b) , (b,c)\}$ be a relation on the set $A=\{a,b,c\}$
not reflexive, not symmetric, & not transitive.

let $R= \{(a,a) , (b,b) , (c,c) , (d,d)\}$ be a relation on the set $A={a,b,c,d}$
reflexive, symmetric, & transitive.
November 23rd 2012, 06:31 AM
Plato

Re: Relations equations i found online and i'm tring to get the answers to understand

Quote:

Originally Posted by math333

2. Suppose R is a symmetric and transitive relation on a setA, and there is an element a ∈ A for which (a; x)∈Rfor all x∈[SIZE=3][COLOR=#000000][FONT=Calibri] A. Prove that R is reflexive.

Suppose that $t\in A$ . We know that $(a,t)\in R$ . WHY?

We know that $(t,a)\in R$ . WHY?

SO?
November 23rd 2012, 06:36 AM
Plato

Re: Relations equations i found online and i'm tring to get the answers to understand

Quote:

Originally Posted by math333

3. Prove or disprove: If a relation is symmetric and transitive, then it isalso reflexive.

let $R= \{(a,a) , (b,b) , (d,d) , (a,b) , (b,a)\}$ be a relation on the set $A=\{a,b,c,d\}$
not reflexive, symmetric, & transitive.