Question

5.How can you proof a proposition in the form of ∀x P(x) is NOT true.

6.a) Briefly explain what does it mean to say B is a subset of A? What is the procedure to prove that?
b) How many subsets of A are there, if |A| = n ?
c) Define an arbitrary set A (with |A|=4), list all the elements of the power set of A. (P(A))

3. Briefly explain how you can prove that two sets are equal.

Answer 1

(5) ∀x P(x) is NOT true means there exists aleast one x for which P(x) is not true.

∃x ~P(x) for example we say that P(x) : x is even and not prime .

Then here ∀x P(x) is not true so we can prove it by saying that there exists 2 which is even but also prime.

(6) (a) B is a subset of A denoted by

It means that every element of B is an element of A.

Take any element belongs to B it must belongs to A if B is a subset of A.

To prove it is take arbitrary element of B and show that it also belongs to A

(b) if |A| then there are subsets of A.

(c) Take A ={a,b,c,d} then

{a}, {b}, {c} , {d} , {a,b} , {a,c} , {a,d} , {b,c} , {b,d} , {c,d} , {a,b,c} ,{b,c,d} , {a,c,d} , {a,b,d} , {a,b,c,d}

(3) If we have to prove that two sets A and B are equal.

Then first we prove that A is subset of B from procedure 6(a)

and then B is a subset of A again from same procedure.

If both are subsets of each other then A=B