Q1. Let 𝐴 = {0,1,2,3,4,5,6,7} suppose 𝑅 𝑎𝑛𝑑 𝑇 are two relations on 𝐴 such that 𝑅 = {(𝑥, 𝑦): 𝑥 + 2𝑦 ≥ 4}, 𝑇 = {(𝑥, 𝑦): 2𝑥 + 3𝑦 ∈ 𝐴} Write 𝑅, 𝑇, 𝑎𝑛𝑑 𝑅°𝑇

First, let's find the pairs for 𝑅 and 𝑇.

For 𝑅 = {(𝑥, 𝑦): 𝑥 + 2𝑦 ≥ 4}, we substitute the values from 𝐴 for 𝑥 and 𝑦 and find the pairs that satisfy the condition.

The pairs for 𝑅 are: {(2,1), (3,1), (4,0), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (4,7), (5,0), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (5,7), (6,0), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (6,7), (7,0), (7,1), (7,2), (7,3), (7,4), (7,5), (7,6), (7,7)}

For 𝑇 = {(𝑥, 𝑦): 2𝑥 + 3𝑦 ∈ 𝐴}, we substitute the values from 𝐴 for 𝑥 and 𝑦 and find the pairs that satisfy the condition.

The pairs for 𝑇 are: {(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)}

Now, let's find 𝑅°𝑇, which is the composition of 𝑅 and 𝑇. A pair (𝑎,𝑏) is in 𝑅°𝑇 if there exists a 𝑐 in 𝐴 such that (𝑎,𝑐) is in 𝑅 and (𝑐,𝑏) is in 𝑇.

The pairs for 𝑅°𝑇 are: {(4,0), (4,1), (5,0), (5,1), (6,0), (6,1), (7,0), (7,1)}