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 pls solve it find R and t

Let's solve this step by step:

Step 1: Define the sets 𝑅 and 𝑇

𝑅 = {(𝑥, 𝑦): 𝑥 + 2𝑦 ≥ 4} is a set of all pairs (x, y) such that x + 2y is greater than or equal to 4.

𝑇 = {(𝑥, 𝑦): 2𝑥 + 3𝑦 ∈ 𝐴} is a set of all pairs (x, y) such that 2x + 3y is an element of the set A.

Step 2: Find the elements of 𝑅 and 𝑇

For 𝑅, we need to find all pairs (x, y) in A x A such that x + 2y ≥ 4. The pairs that satisfy this condition are: (2,1), (3,1), (3,2), (3,3), (4,0), (4,1), (4,2), (4,3), (5,0), (5,1), (5,2), (5,3), (6,0), (6,1), (6,2), (6,3), (7,0), (7,1), (7,2), (7,3).

For 𝑇, we need to find all pairs (x, y) in A x A such that 2x + 3y is an element of A. The pairs that satisfy this condition are: (0,0), (0,1), (0,2), (1,0), (1,1), (2,0), (2,1), (3,0), (3,1), (4,0).

Step 3: Find 𝑅°𝑇

The composition of two relations, 𝑅°𝑇, is a set of all pairs (x, y) such that there exists a z in A for which (x, z) is in 𝑇 and (z, y) is in 𝑅.

To find 𝑅°𝑇, we need to check all possible combinations of pairs in 𝑇 and 𝑅. If the second element of a pair from 𝑇 is the same as the first element of a pair from 𝑅, then we form a new pair where the first element is the first element of the pair from 𝑇 and the second element is the second element of the pair from 𝑅.

After checking all combinations, we find that 𝑅°𝑇 = {(0,1), (0,2), (0,3), (1,1), (1,2), (1,3), (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2), (3,3), (4,0)}.