(c) Let A, B, and C be subsets of some universal set. If A ¢ B and BC C, then A £C.
(c) Let A, B, and C be subsets of some universal set. If A ¢ B and BC C, then A £C.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![17. Evaluation of Proofs
See the instructions for Exercise (19) on page 100 from Section 3.1.
(a) Let A, B, and C be subsets of some universal set. If A ¢ B and
B¢ C, then A 4 C.
Proof. We assume that A, B, and C are subsets of some universal set
and that A Z B and B ¢ C. This means that there exists an element x
in A that is not in B and there exists an element x that is in B and not in
C. Therefore, x e A and x ¢ C, and we have proved that A ¢ C. I
(b) Let A, B, and C be subsets of some universal set. If AO B = AN C,
then B = C.
Proof. We assume that A N B = A n C and will prove that B = C.
We will first prove that B C C.
So let x e B. If x € A, then x e A O B, and hence, x e ANC.
From this we can conclude that x e C. If x ¢ A, then x ¢ AN B,
and hence, x ¢ ANC. However, since x ¢ A, we may conclude that
хеС. Therefore, в С.
The proof that C C B may be done in a similar manner. Hence, B =
С.
(c) Let A, B, and C be subsets of some universal set. If A ¢ B and
BC C, then A g C.
Proof. Assume that A 4 B and BC C. Since A ¢ B, there exists
an element x such that x e A and x ¢ B. Šince B C Ĉ, we may
conclude that x ¢ C. Hence, x e A and x ¢ C, and we have proved
that A 4 C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ce9e3a3-dedc-4902-85ce-d3b117f9e3ef%2F27926f05-f8ca-4bee-8843-67b36c25deae%2Fmq1dfnp_processed.png&w=3840&q=75)
Transcribed Image Text:17. Evaluation of Proofs
See the instructions for Exercise (19) on page 100 from Section 3.1.
(a) Let A, B, and C be subsets of some universal set. If A ¢ B and
B¢ C, then A 4 C.
Proof. We assume that A, B, and C are subsets of some universal set
and that A Z B and B ¢ C. This means that there exists an element x
in A that is not in B and there exists an element x that is in B and not in
C. Therefore, x e A and x ¢ C, and we have proved that A ¢ C. I
(b) Let A, B, and C be subsets of some universal set. If AO B = AN C,
then B = C.
Proof. We assume that A N B = A n C and will prove that B = C.
We will first prove that B C C.
So let x e B. If x € A, then x e A O B, and hence, x e ANC.
From this we can conclude that x e C. If x ¢ A, then x ¢ AN B,
and hence, x ¢ ANC. However, since x ¢ A, we may conclude that
хеС. Therefore, в С.
The proof that C C B may be done in a similar manner. Hence, B =
С.
(c) Let A, B, and C be subsets of some universal set. If A ¢ B and
BC C, then A g C.
Proof. Assume that A 4 B and BC C. Since A ¢ B, there exists
an element x such that x e A and x ¢ B. Šince B C Ĉ, we may
conclude that x ¢ C. Hence, x e A and x ¢ C, and we have proved
that A 4 C.
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