Let n be a positive integer and let X = {1, 2,...,n}. Suppose that R is...

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Let n be a positive integer and let X = {1, 2,...,n}. Suppose that R is a symmetric and transitive relation on X satisfying the following condition: In the matrix of the relation R (relative to the ordering 1,2,...,n), every row has at least one nonzero entry. Order 7 of the following sentences so that they form a proof for the following statment: R is an equivalence relation on X. Choose from these sentences: Since R is symmetric and transitive, in order to show that R is an equivalence relation, we only need to show that it is reflexive. Therefore R is reflexive. Since (z,y), (y, z) = R and R is transitive, we have (x,x) = R. Since (x, y) = R and R is symmetric, we have (y, z) € R. Since the matrix of relation R has a nonzero entry in the row corresponding to z, there is at least one element y € X such that (x, y) = R. Suppose (x, y), (y,x) € R. For an arbitrary x € X, we have shown that (1, 1) E R. Since R is symmetric and transitive, in order to show that R is an equivalence relation, we must show that it is reflexive and not antisymmetric. Therefore R is not antisymmetric. Suppose à € X. Your Proof: Let n be a positive integer and let X = {1, 2,...,n}. Suppose that R is a symmetric and transitive relation on X satisfying the following condition: In the matrix of the relation R (relative to the ordering 1,2,...,n), every row has at least one nonzero entry. Order 7 of the following sentences so that they form a proof for the following statment: R is an equivalence relation on X. Choose from these sentences: Since R is symmetric and transitive, in order to show that R is an equivalence relation, we only need to show that it is reflexive. Therefore R is reflexive. Since (z,y), (y, z) = R and R is transitive, we have (x,x) = R. Since (x, y) = R and R is symmetric, we have (y, z) € R. Since the matrix of relation R has a nonzero entry in the row corresponding to z, there is at least one element y € X such that (x, y) = R. Suppose (x, y), (y,x) € R. For an arbitrary x € X, we have shown that (1, 1) E R. Since R is symmetric and transitive, in order to show that R is an equivalence relation, we must show that it is reflexive and not antisymmetric. Therefore R is not antisymmetric. Suppose à € X. Your Proof:

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Solution To show is equivalence relation on X Step 1 ... View the full answer