Discrete Mathematics

[Mo Xiang]

1. The relation R on ? = {−2,−1,1,2} is defined by ??? if and only if ?3 + ?3 > 0.
a. List the ordered pairs that belong to the relation R.
b. Find the in-degree and out-degree of each vertex.
c. State the domain and range of the relation R.
d. Determine whether the relation R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric or transitive. Give a counterexample if the answer is ‘No’.
e. Determine whether R is an equivalence relation. Justify your answer.
f. Find the reflexive closure of R. Write your answer in matrix form.

2. Let x Î {1, 2} and y Î {1, 4}, consider a predicate P(x, y) defined as P(x, y): 4 divides (?2 + 3?). Rewrite the expression $y["xP(x, y)] by eliminating the quantifiers and replace P(x, y) with 4 divides (?2 + 3?). Hence, determine the truth value of the statement.

 

[Deleted member 4993]

1. The relation R on ? = {−2,−1,1,2} is defined by ??? if and only if ?3 + ?3 > 0.
a. List the ordered pairs that belong to the relation R.
b. Find the in-degree and out-degree of each vertex.
c. State the domain and range of the relation R.
d. Determine whether the relation R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric or transitive. Give a counterexample if the answer is ‘No’.
e. Determine whether R is an equivalence relation. Justify your answer.
f. Find the reflexive closure of R. Write your answer in matrix form.

2. Let x Î {1, 2} and y Î {1, 4}, consider a predicate P(x, y) defined as P(x, y): 4 divides (?2 + 3?). Rewrite the expression $y["xP(x, y)] by eliminating the quantifiers and replace P(x, y) with 4 divides (?2 + 3?). Hence, determine the truth value of the statement.

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