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**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.