Question

The domain of discourse is sets. Suppose some predicate Q is true for all elemenst of A: ∀a ∈ A Q(a). Recall the definition of subset: X ⊆ Y is defined as ∀b ∈ X b ∈ Y . Suppose B ⊆ A and C ⊆ B. Prove that ∀c ∈ C Q(c)

Answer 1

Answer:

See the step-by-step explanation

Step-by-step explanation:

Let c be any element of C. (I'm not sure wether you have to assume that C is non-empt or not)

C is a subset of B. That means that as c is  in C, it is also in B. ($c \in C \Rightarrow c \in B$)

Now, B is a subset of A. It follows that as $c \in B \Rightarrow c \in A$.

That means c is an element of A. The predicate Q is true for all elements of A, including c.

Because we let c be any element of C, we have proven that the predicate Q is true for all elements in C.