Given the sets A=[a,b], B=[a,b,c], C=[b,c,d]
Prove that if A is a subset of B and
B is a subset of C
then A is a subset of C
To prove that if A is a subset of B and B is a subset of C, then A is a subset of C, we can use the definition of subset and set inclusion.
According to the definition, set A is a subset of set B if every element of A is also an element of B. Similarly, set B is a subset of set C if every element of B is also an element of C.
Given that A=[a,b], B=[a,b,c], and C=[b,c,d], let's analyze whether A is a subset of B and B is a subset of C.
1. A is a subset of B:
Since A=[a,b] and B=[a,b,c], every element of A (a and b) is also found in B. Therefore, A is a subset of B.
2. B is a subset of C:
Similarly, since B=[a,b,c] and C=[b,c,d], every element of B (a, b, and c) is also found in C. Hence, B is a subset of C.
Now, to prove that A is a subset of C, we need to show that every element in A is also present in C.
In set A [a,b], both a and b are elements of B and B is a subset of C. Therefore, a and b are also elements of C. Thus, every element of A is present in C.
Since every element of A is found in C, we can conclude that A is a subset of C.
In summary, if A is a subset of B and B is a subset of C, then it follows that A is a subset of C.