Show that the relation R in the set {1,2,3} given by R={(1,1),(2,2), (3,3),(1,2),(2,3)} is reflexive but neither symmetric nor transitive.
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Step-by-Step Solution
Step 1: Check for Reflexivity
A relation R on a set A is reflexive if every element in A is related to itself. This means that for every element a in the set A, the ordered pair (a,a) must be present in the relation R. The given set is A={1,2,3}. We need to check if (1,1), (2,2), and (3,3) are in R.
Step 2: Check for Symmetry
A relation R is symmetric if, whenever an element a is related to an element b, then b is also related to a. In other words, if (a,b) is in R, then (b,a) must also be in R. We will check the pairs in R to see if their reverse pairs are also present.
Step 3: Check for Transitivity
A relation R is transitive if, whenever a is related to b and b is related to c, then a is also related to c. This means if (a,b) and (b,c) are in R, then (a,c) must also be in R. We will examine the pairs in R to verify this condition.