Q1. Let ๐ด = {0,1,2,3,4,5,6,7} suppose ๐ ๐๐๐ ๐ are two relations on ๐ด such that ๐ = {(๐ฅ, ๐ฆ): ๐ฅ + 2๐ฆ โฅ 4}, ๐ = {(๐ฅ, ๐ฆ): 2๐ฅ + 3๐ฆ โ ๐ด} Write ๐ , ๐, ๐๐๐ ๐ ยฐ๐
Question
Q1. Let ๐ด = {0,1,2,3,4,5,6,7} suppose ๐ and ๐ are two relations on ๐ด such that
๐
= {(๐ฅ, ๐ฆ): ๐ฅ + 2๐ฆ โฅ 4},
๐ = {(๐ฅ, ๐ฆ): 2๐ฅ + 3๐ฆ โ ๐ด}
Write ๐
, ๐, and ๐
ยฐ๐.
Solution
First, let's find the pairs for ๐ and ๐.
For ๐ = {(๐ฅ, ๐ฆ): ๐ฅ + 2๐ฆ โฅ 4}, we substitute the values from ๐ด for ๐ฅ and ๐ฆ and find the pairs that satisfy the condition.
The pairs for ๐ 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 ๐ = {(๐ฅ, ๐ฆ): 2๐ฅ + 3๐ฆ โ ๐ด}, we substitute the values from ๐ด for ๐ฅ and ๐ฆ and find the pairs that satisfy the condition.
The pairs for ๐ are: {(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)}
Now, let's find ๐ ยฐ๐, which is the composition of ๐ and ๐. A pair (๐,๐) is in ๐ ยฐ๐ if there exists a ๐ in ๐ด such that (๐,๐) is in ๐ and (๐,๐) is in ๐.
The pairs for ๐ ยฐ๐ are: {(4,0), (4,1), (5,0), (5,1), (6,0), (6,1), (7,0), (7,1)}
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