Q1. Consider the relation R on the set of integers as xRy if and only if x<y. Then prove that R is partial order relation.

Define Partial Order relation and check whether R is Partial Order relation.R= {(x,y) 𝑖𝑓 𝑦 = 𝑥𝑟, 𝑟 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 𝑎, 𝑏 ∈ 𝑍}

Consider the relation R on P dened by:R = (x; y) | y = 2^i * x for some i 2 N:(a) Prove that R is a partial order on N.(b) Identify the minimal elements of R.(c) Let A = {1; 2; 3; 5; 6; 12}. Draw the Hasse diagram for the poset dened by this relation

Assume the R is a relation on a set A, aRb is partially ordered such that a and b

Let R be the relation on the set Z defined by xRy iff x − y is an integer. Prove that R is anequivalence relation on Z.

Let R be the relation on Z≥ (the set of integers) defined by (x, y) ∈ R iff x2 + y2 = 2k for some integers k ≥0.Which one of the following is an ordered pair in R?a.(1, 0)b.(2, 9)c.(3, 8)d.(5, 7)