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#mathematics
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#relations
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5, Let \( R \) is arelation on the ret \( A=\{1,2,3,4 \) \( 5,6\} \) detined b4 \( R=4(a,+b) ; a+b <=9\).
i) Lest the element of \( R \)
ii) is \( R=R^{-1} \).?

Mar 14

5, Let \( R \) is arelation on the ret \( A=\{1,2,3,4 \) \( 5,6\} \) detined b4 \( R=4(a,+b) ; a+b <=9\).

i) Lest the element of \( R \)

ii) is \( R=R^{-1} \).

i) Lest the element of \( R \)

ii) is \( R=R^{-1} \).

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ChatGPT

Mar 14

i) The elements of \( R \) can be found by plugging in all possible pairs of elements from set \( A \) into the relation \( R=4(a+b) \) where \( a+b \leq 9 \).

So, the elements of \( R \) are:

\( R = \{4(1+1), 4(1+2), 4(1+3), 4(1+4), 4(1+5), 4(1+6), 4(2+2), 4(2+3), 4(2+4), 4(2+5), 4(2+6), 4(3+3), 4(3+4), 4(3+5), 4(3+6), 4(4+4), 4(4+5), 4(4+6), 4(5+5), 4(5+6), 4(6+6)\}

\( R = \{8, 12, 16, 20, 24, 28, 16, 20, 24, 28, 32, 24, 28, 32, 36, 32, 36, 40, 40, 44, 48\} \)

ii) To check if \( R = R^{-1} \), we need to find the inverse of \( R \). The inverse of a relation \( R \) is the set of all pairs \( (b, a) \) such that \( (a, b) \) is in \( R \).

So, the inverse of \( R \) is:

\( R^{-1} = \{(8, 1), (12, 1), (16, 1), (20, 1), (24, 1), (28, 1), (16, 2), (20, 2), (24, 2), (28, 2), (32, 2), (24, 3), (28, 3), (32, 3), (36, 3), (32, 4), (36, 4), (40, 4), (40, 5), (44, 5), (48, 6)\} \)

Since \( R \) and \( R^{-1} \) are not equal, \( R \neq R^{-1} \).

So, the elements of \( R \) are:

\( R = \{4(1+1), 4(1+2), 4(1+3), 4(1+4), 4(1+5), 4(1+6), 4(2+2), 4(2+3), 4(2+4), 4(2+5), 4(2+6), 4(3+3), 4(3+4), 4(3+5), 4(3+6), 4(4+4), 4(4+5), 4(4+6), 4(5+5), 4(5+6), 4(6+6)\}

\( R = \{8, 12, 16, 20, 24, 28, 16, 20, 24, 28, 32, 24, 28, 32, 36, 32, 36, 40, 40, 44, 48\} \)

ii) To check if \( R = R^{-1} \), we need to find the inverse of \( R \). The inverse of a relation \( R \) is the set of all pairs \( (b, a) \) such that \( (a, b) \) is in \( R \).

So, the inverse of \( R \) is:

\( R^{-1} = \{(8, 1), (12, 1), (16, 1), (20, 1), (24, 1), (28, 1), (16, 2), (20, 2), (24, 2), (28, 2), (32, 2), (24, 3), (28, 3), (32, 3), (36, 3), (32, 4), (36, 4), (40, 4), (40, 5), (44, 5), (48, 6)\} \)

Since \( R \) and \( R^{-1} \) are not equal, \( R \neq R^{-1} \).