Recall that a relation R from A to B is a subset RCA x B. (So for example, a relation on a set X is a relation from X to X.) Just as we can define new sets from old ones using "set operations" (e.g., n, U, x, etc.) we can also define new relations from old ones. These give rise to what is known as relational algebra. In what follows, use the following sets X, Y, Z: X = {1,2,3,4} Y = {a,b,c,d} Z= {ant, bat, cat} and define relations S from X to Y, and T from Y to Z as follows: S = {(1, a), (1, c), (2, a), (2, c), (2, d), (4, a)} T = {(a, ant), (b, bat), (c, cat)} (a) For a set A, the identity relation on A is the relation on A defined by IA = {(a, a) | a € A}. Write down Ix in roster form. (b) Given a relation R from A to B, the inverse of R is the relation from B to A defined by R-¹ = {(b, a) | (a, b) ≤ R}. Write down S-1 in roster form. (c) Suppose that R₁ is a relation from A to B, and that R₂ is a relation from B to C. The composition of R₁ and R₂ is the relation from A to C defined by R₂0 R₁ = {(a, c) | (‡b ≤ B) ((a, b) ≤ R₁ ^ (b, c) € R₂)}. 1) Write down So Ix in roster form. What do you observe? 2) Write down Iy o S in roster form. What do you observe? 3) Explain why the expression "Ixo S" does not make sense.
Recall that a relation R from A to B is a subset RCA x B. (So for example, a relation on a set X is a relation from X to X.) Just as we can define new sets from old ones using "set operations" (e.g., n, U, x, etc.) we can also define new relations from old ones. These give rise to what is known as relational algebra. In what follows, use the following sets X, Y, Z: X = {1,2,3,4} Y = {a,b,c,d} Z= {ant, bat, cat} and define relations S from X to Y, and T from Y to Z as follows: S = {(1, a), (1, c), (2, a), (2, c), (2, d), (4, a)} T = {(a, ant), (b, bat), (c, cat)} (a) For a set A, the identity relation on A is the relation on A defined by IA = {(a, a) | a € A}. Write down Ix in roster form. (b) Given a relation R from A to B, the inverse of R is the relation from B to A defined by R-¹ = {(b, a) | (a, b) ≤ R}. Write down S-1 in roster form. (c) Suppose that R₁ is a relation from A to B, and that R₂ is a relation from B to C. The composition of R₁ and R₂ is the relation from A to C defined by R₂0 R₁ = {(a, c) | (‡b ≤ B) ((a, b) ≤ R₁ ^ (b, c) € R₂)}. 1) Write down So Ix in roster form. What do you observe? 2) Write down Iy o S in roster form. What do you observe? 3) Explain why the expression "Ixo S" does not make sense.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Recall that a relation R from A to B is a subset RCA × B. (So for example, a relation on
a set X is a relation from X to X.) Just as we can define new sets from old ones using "set
operations” (e.g., N, U, ×, etc.) we can also define new relations from old ones. These give
rise to what is known as relational algebra.
In what follows, use the following sets X, Y, Z:
X = {1,2,3,4}
Y = {a, b, c, d}
Z = {ant, bat, cat}
and define relations S from X to Y, and T from Y to Z as follows:
S = {(1, a), (1, c), (2, a), (2, c), (2, d), (4, a)}
T = {(a, ant), (b, bat), (c, cat)}
(a) For a set A, the identity relation on A is the relation on A defined by
IA = {(a, a) | a € A} .
Write down Ix in roster form.
(b) Given a relation R from A to B, the inverse of R is the relation from B to A defined
by
R−¹ = {(b, a) | (a, b) ≤ R} .
Write down S-1 in roster form.
(c) Suppose that R₁ is a relation from A to B, and that R₂ is a relation from B to C. The
composition of R₁ and R₂ is the relation from A to C defined by
R₂ ° R₁ = {(a, c) | (‡b € B)((a, b) ≤ R₁ ^ (b, c) = R₂)}.
1) Write down So Ix in roster form. What do you observe?
2) Write down Iy o S in roster form. What do you observe?
3) Explain why the expression "Ix o S" does not make sense.
4) Write down To T-¹ in roster form. What do you observe? Based on your prior
experience with similar notation, is this what you expected?
5) Write down T-¹ oT in roster form. What do you observe? Is anything “missing”?
Based on your prior experience with similar notation, is this what you expected?
6) Write down To S in roster form.
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