Proving Power Set Inclusion When B is Included in C

AI Thread Summary
To prove that if set B is included in set C, then the power set of B is included in the power set of C, one can start with the definitions of power sets and subsets. The power set of C contains all subsets of C, which inherently includes all subsets of any subset of C, including B. Therefore, every subset of B, which constitutes the power set of B, is also a subset of C. This leads to the conclusion that the power set of B must be included in the power set of C. The argument is formal and aligns with set theory principles.
yxgao
Messages
122
Reaction score
0
How do you show that if B is included in C, then the power set of B is included in the power set of C?
 
Physics news on Phys.org
What have you tried already? I'd imagine just starting with the definitions would give a strong clue.
 
Power set of C includes all subsets of C, which means that it also includes all subsets of a subset (B). Thus it includes the power set of B.
Is this formal enough?
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
View full post »

Similar threads

I Countability of binary Strings
Replies
18
Views
2K
MHB ZFC and the Axiom of Power Sets ....
Replies
2
Views
2K
I Cardinality of the Power Series of an Infinite Set
Replies
19
Views
4K
MHB Translate the statements into set inclusion
Replies
5
Views
1K
MHB Operations on Sets: Proving A⊆B⊆C & A∪B=B∩C
Replies
2
Views
2K
I Axiom of Choice: Disjoint Family ##\Rightarrow ## Power Set
Replies
10
Views
3K
Is the Subtraction of Power Sets Possible?
Replies
4
Views
6K
I Thinking about equality of infinite sets
Replies
5
Views
2K
MHB Proof of a set union and intersection
Replies
1
Views
1K
MHB Topology Munkres Chapter 1 exercise 2 b and c- Set theory equivalent statements
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top