let A and B be subsets of a universal set. Show that A subset of B if and only if B' subset of A'.
To show that A subset of B if and only if B' subset of A':
First, assume that A is a subset of B. That is, every element of A is also an element of B.
Now, we want to show that B' is a subset of A'. To prove this, we need to show that every element in B' is also in A'.
Let x be an arbitrary element of B'. This means that x is not in B. But since A is a subset of B, this also implies that x is not in A. Hence, x must belong to A', the complement of A.
Therefore, we have shown that B' is a subset of A'.
Next, assume that B' is a subset of A'. That is, every element of B' is also an element of A'.
Now, we want to show that A is a subset of B. To prove this, we need to show that every element in A is also in B.
Let y be an arbitrary element of A. This means that y is not in A'. But since B' is a subset of A', this also implies that y is not in B'. Hence, y must belong to B.
Therefore, we have shown that A is a subset of B.
Hence, we have proved that A subset of B if and only if B' subset of A'.