So, you have 50 questions in one thread (5 relations times 10 categories), and you don't make any attempt to answer them or even say what difficulty you are having?
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.
Help me please