(1)Given the sets A={a,b}, B={a,b,c},
C= {b,c,d}. which of these sets are:
(i) Equal
(ii) Comparable
(iii) Subset
(2) Prove that if A is a subset of B and B is a subset of C then A is a subset of C
What is your thinking on these?
To determine the relationships between sets and whether they are equal, comparable, or subsets, we can follow these guidelines:
1. If two sets have the same elements, they are considered equal.
2. Two sets are considered comparable if every element in the first set is also in the second set, and vice versa.
3. A set A is considered a subset of a set B if every element in A is also an element in B.
Using these guidelines, let's analyze the given sets:
(1) Given sets:
A = {a, b}
B = {a, b, c}
C = {b, c, d}
(i) To check for equality, compare each set to one another:
- A and B: A does not contain the element c, so A is not equal to B.
- A and C: A does not contain the element d, so A is not equal to C.
- B and C: B does not contain the element d, so B is not equal to C.
Therefore, none of the sets A, B, and C are equal to each other.
(ii) To check for comparability, compare each set to one another:
- A and B: Every element in A (a, b) is also present in B (a, b, c). Similarly, every element in B is present in A. Thus, A and B are comparable.
- A and C: A and C share the element b, but A does not contain c and C does not contain a. Therefore, A and C are not comparable.
- B and C: B and C share the elements b and c, but B does not contain d, and C does not contain a. Hence, B and C are not comparable.
So, only A and B are comparable.
(iii) To check for subsets, compare each set to one another:
- A is a subset of B if every element in A is also in B. In this case, every element in A (a, b) is also present in B (a, b, c). Thus, A is a subset of B.
- A is a subset of C if every element in A is also in C. In this case, C does not contain the element a, so A is not a subset of C.
- B is a subset of C if every element in B is also in C. In this case, every element in B (a, b, c) is also present in C (b, c, d). Thus, B is a subset of C.
Therefore, A is a subset of B, and B is a subset of C.
(2) To prove that if A is a subset of B and B is a subset of C, then A is a subset of C:
Let's assume that A is a subset of B and B is a subset of C. We want to prove that A is a subset of C.
Assume x is an arbitrary element in A. Since A is a subset of B, x must also be an element of B.
Now, since B is a subset of C, x must also be an element of C.
Since x is an arbitrary element in A and x is also an element of C, we can conclude that every element in A is also an element of C.
Therefore, A is a subset of C.
Thus, we have proven that if A is a subset of B and B is a subset of C, then A is a subset of C.