Sure, let's delve deep into these propositions that belong to the field of set theory:
1) (A union B) subtract (B subtract A) = (A union B) subtract B;
Let's break this down. Firstly, we look at (B subtract A). This consists of only the elements that are present in B but not in A.
Next, when this is subtracted from (A union B), which will be all the elements present in either A or B, the remaining elements will be those only in A. We can visualize this as a Venn diagram where only the section representing A remains. This confirms the truth of the proposition; that the left side is equal to the right side: (A union B) subtract B.
2) If A is subset of B, then (A intersect C) is subset of (B intersect C);
A set A is said to be a subset of set B if all elements of A are also elements of B. Now consider a set C.
The intersection of A and C (A intersect C) would contain elements that are both in A and C. If A is a subset of B, then these common elements would also exist in B as well. Therefore, the set (A intersect C) should be a subset of (B intersect C). It simply means that every element that is both in A and C is also in both B and C, proving the proposition to be true.
So, according to set theory, both these propositions stand true.