For each of the following relations defined on sets M = {5, 6, 7, 8} and N = {1, 3, 5, 7, 9},list the elements and find its domain and range.(a) S1 = {(m, n)|m + n is even}(b) S2 = {(m, n)|n = 2m − 9}(c) S3 = {(m, n)|m < n}(d) S4 = {(m, n)||m − n| ≤ 2}(e) S5 = {(m, n)|m is divisible by n}(f) S6 = {(m, n)|m + n is a prime number}
Question
For each of the following relations defined on sets M = {5, 6, 7, 8} and N = {1, 3, 5, 7, 9},list the elements and find its domain and range.(a) S1 = {(m, n)|m + n is even}(b) S2 = {(m, n)|n = 2m − 9}(c) S3 = {(m, n)|m < n}(d) S4 = {(m, n)||m − n| ≤ 2}(e) S5 = {(m, n)|m is divisible by n}(f) S6 = {(m, n)|m + n is a prime number}
Solution
(a) S1 = {(m, n) | m + n is even}
Para determinar los elementos de S1, verificamos cuáles pares (m, n) cumplen con la condición de que m + n sea par.
- (5, 1) → 5 + 1 = 6 (par)
- (5, 3) → 5 + 3 = 8 (par)
- (6, 5) → 6 + 5 = 11 (impar)
- (6, 7) → 6 + 7 = 13 (impar)
- (7, 1) → 7 + 1 = 8 (par)
- (7, 3) → 7 + 3 = 10 (par)
- (8, 5) → 8 + 5 = 13 (impar)
- (8, 7) → 8 + 7 = 15 (impar)
- (5, 5) → 5 + 5 = 10 (par)
- (5, 7) → 5 + 7 = 12 (par)
- (6, 1) → 6 + 1 = 7 (impar)
- (6, 3) → 6 + 3 = 9 (impar)
- (7, 5) → 7 + 5 = 12 (par)
- (7, 7) → 7 + 7 = 14 (par)
- (8, 1) → 8 + 1 = 9 (impar)
- (8, 3) → 8 + 3 = 11 (impar)
- (5, 9) → 5 + 9 = 14 (par)
- (6, 9) → 6 + 9 = 15 (impar)
- (7, 9) → 7 + 9 = 16 (par)
- (8, 9) → 8 + 9 = 17 (impar)
Elementos de S1: {(5, 1), (5, 3), (5, 5), (5, 7), (5, 9), (7, 1), (7, 3), (7, 5), (7, 7), (7, 9)}
Dominio de S1: {5, 7} Rango de S1: {1, 3, 5, 7, 9}
(b) S2 = {(m, n) | n = 2m − 9}
Para determinar los elementos de S2, verificamos cuáles pares (m, n) cumplen con la condición de que n = 2m − 9.
- (5, 1) → 2(5) - 9 = 10 - 9 = 1
- (6, 3) → 2(6) - 9 = 12 - 9 = 3
- (7, 5) → 2(7) - 9 = 14 - 9 = 5
- (8, 7) → 2(8) - 9 = 16 - 9 = 7
Elementos de S2: {(5, 1), (6, 3), (7, 5), (8, 7)}
Dominio de S2: {5, 6, 7, 8} Rango de S2: {1, 3, 5, 7}
(c) S3 = {(m, n) | m < n}
Para determinar los elementos de S3, verificamos cuáles pares (m, n) cumplen con la condición de que m < n.
- (5, 7)
- (5, 9)
- (6, 7)
- (6, 9)
- (7, 9)
Elementos de S3: {(5, 7), (5, 9), (6, 7), (6, 9), (7, 9)}
Dominio de S3: {5, 6, 7} Rango de S3: {7, 9}
(d) S4 = {(m, n) | |m − n| ≤ 2}
Para determinar los elementos de S4, verificamos cuáles pares (m, n) cumplen con la condición de que |m − n| ≤ 2.
- (5, 3) → |5 - 3| = 2
- (5, 5) → |5 - 5| = 0
- (5, 7) → |5 - 7| = 2
- (6, 5) → |6 - 5| = 1
- (6, 7) → |6 - 7| = 1
- (7, 5) → |7 - 5| = 2
- (7, 7) → |7 - 7| = 0
- (7, 9) → |7 - 9| = 2
- (8, 7) → |8 - 7| = 1
- (8, 9) → |8 - 9| = 1
Elementos de S4: {(5, 3), (5, 5), (5, 7), (6, 5), (6, 7), (7, 5), (7, 7), (7, 9), (8, 7), (8, 9)}
Dominio de S4: {5, 6, 7, 8} Rango de S4: {3, 5, 7, 9}
(e) S5 = {(m, n) | m is divisible by n}
Para determinar los elementos de S5, verificamos cuáles pares (m, n) cumplen con la condición de que m es divisible por n.
- (5, 1) → 5 es divisible por 1
- (6, 1) → 6 es divisible por 1
- (7, 1) → 7 es divisible por 1
- (8, 1) → 8 es divisible por 1
- (6, 3) → 6 es divisible por 3
Elementos de S5: {(5, 1), (6, 1), (7, 1), (8, 1), (6, 3)}
Dominio de S5: {5, 6, 7, 8} Rango de S5: {1, 3}
(f) S6 = {(m, n) | m + n is a prime number}
Para determinar los elementos de S6, verificamos cuáles pares (m, n) cumplen con la condición de que m + n sea un número primo.
- (5, 2) → 5 + 2 = 7 (primo)
- (5, 4) → 5 + 4 = 9 (no primo)
- (5, 6) → 5 + 6 = 11 (primo)
- (5, 8) → 5 + 8 = 13 (primo)
- (6, 1) → 6 + 1 = 7 (primo)
- (6, 5) → 6 + 5 = 11 (primo)
- (7, 2) → 7 + 2 = 9 (no primo)
- (7, 4) → 7 + 4 = 11 (primo)
- (7, 6) → 7 + 6 = 13 (primo)
- (8, 1) → 8 + 1 = 9 (no primo)
- (8, 3) → 8 + 3 = 11 (primo)
- (8, 5) → 8 + 5 = 13 (primo)
Elementos de S6: {(5, 2), (5, 6), (5, 8), (6, 1), (6, 5), (7, 4), (7, 6), (8, 3), (8, 5)}
Dominio de S6: {5, 6, 7, 8} Rango de S6: {1, 2, 3, 4, 5, 6, 8}
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