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x=0 [ x ] = R b. x R y ⇔ x > y ∀ x, y ϵ R Not reflexive: A
counterexample to prove is not would be x=y, x=4; therefore, x
should be greater than y, but since 4 is not greater than 4, this
relationship is not
reflexive. Not Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. For any value of x and y if (x,y) belongs to R
i.e, x>y
then y>x is not possible so we can say that R is not symmetric
There is no x, y pairs that relate back to each other. E.g. (x, y) is
found, but not
(y, x) for all x, y in R. Therefore, it is antisymmetric. Transitive - x, y,
z are related. If x > y, and y > z, then x > z. Is not an
equivalence relation, nor partial order.
c. x R y ⇔ |x| = |y| ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R. If we consider
any value of x then |x|=|x| will hold. R is reflexive Symmetric - It is
symmetric was for all (x, y) there is a corresponding (y, x) pair.
E.g. (-1, 1), (1, -1). Because it is symmetric cannot be antisymmetric.
Not transitive since no number is related to each other. Is not an
equivalence relation, nor partial order. 5. (10 pts) Determine whether
the following pair of statements are logically
equivalent. Justify your answer using a truth table.
p → (q → r) and p ∧ q → r p ∧ q → r p (q r)
p Q r T
T
T