# Thread: Mathematical logic--> Binary relations, orders, and equivalence relations/classes

1. ## Mathematical logic--> Binary relations, orders, and equivalence relations/classes

For each of the following binary relations on the set of natural numbers
including 0, state (yes/no) whether it is (i) reflexive, (ii) irreflexive, (iii)
symmetric, (iv) antisymmetric and/or (v) transitive. Briefly justify your
answers. Identify which relations (if any) are partial orders, strict partial
orders, total orders, and strict total orders. Identify which are equivalence
relations, and give a brief description of the equivalence classes in those
cases.

(a) R1, where mR1n if and only if one of the following holds:
i. m and n are both even and m < n;
ii. m and n are both odd and n < m; or
iii. m is odd and n is even.

(b) R2 where mR2n if and only if m and n have the same number of
distinct prime factors. (1 has no prime factors, and 0 has every
prime as a factor; 12 = 2 • 2 • 3 has 2 distinct prime factors.)

(c) R3, where mR3n if and only if there is a prime p such that m+n = 2p.

(d) R4, where mR4n if and only if 3∣(m + n²).

(e) R5, where mR5n if and only if∣m - n∣ is divisible by both 6 and 10.