## Chapter - 2, Quantificational Logic

### Section - 2.3 - More Operations on Sets

### Summary

- Sets can be defined in following ways:
- Listing all elements of the set. Eg: .
- Element-hood Test Notation: . Or sometimes replacing by an expression eg: .
- A set can also be defined using
*indexed family*notation. For eg: , where .

- An indexed family can also be defined as . Thus means the same thing as .
- Sets whose elements are all sets are called families of sets. Eg: .
- Power Set: .
- If is a family of sets, then:
- Intersection of all sets in .
- Union of all sets in .

- Some useful logical forms:
- is equivalent to .
- is equivalent to which is equivalent to .
- is equivalent to , or equivalently,.
- is equivalent to .

- An alternative notation of union or intersection. If , where each is a set. Then would be the set of all elements common to all the ’s, for , and this can also be written as .

**Soln1**

**(a)**

**(b)**

**(c)**

This is equivalent to:

.

**(d)**

**Soln2**

**(a)**

**(b)**

**(c)**

**(d)**

**Soln3**

**Soln4**

**Soln5**

**Soln6**

Given: where

**Soln8**

Given: and where

**(a)**

**(b)**

**(c)** No. The given statements are not equivalent.

**Soln9**

Thus

**Soln10**

Hence Proved.

**Soln11**

, and

And

Clearly

**Soln12**

**(a)**

Existential quantifier distributes over interjection:

This is equivalent to RHS.

**(b)**

Using reverse of, Universal Quantifier distributes over conjunction:

Using inverse of law of distribution:

Using Demorgan’s Law:

= RHS.

**(c)**

= RHS.

**Soln13**

Given: where and

**(a)**

**(b)**

Putting values from (a) :

**(c)**

To find , lets first compute

Thus we have:

No, expression of (b) and (c) parts are not equal.

**(d)**

LHS:

RHS:

Clearly LHS and RHS are not equal. We already saw this that if order of quantifiers are changed, expressions are also changed.

**Soln14**

**(a)**

If , will always be false.

Thus whole statement will be false.
Thus , irrespective of the value of x.

which means .

**(b)**

If , will always be false.

Thus will always be true.

That means , irrespective of the value of x.

That means .

**Soln15**

**(a)**

- According to the definition, R is the set of all sets that does not contains themselves.
- R itself should also be a set that does not contain itself, as if it contains itself it will contradict its own definition.
- Now if R does not contain itself, then again it contradicts with its definition which says it is a set that contains all the sets that does not contain themselves.
- Thus such set R can not exist.

**(b)** It follows that there is no universal set of sets.