## Chapter - 1, Sentential Logic

### Section - 1.3 - Variable and Sets

### Summary

- Statements with Variables. For eg: “x is divisible by 9”, “y is a person” are statements. Here x, and y are variables.
- These statements are true or false based on the value of variables.
- Sets, a collection of objects.
- Bound and Unbound variables. Eg: , is a free variable, whereas is a bound variable.
- Free variables in a statement are for objects for which statement is talking about.
- Bound variables are just dummy variables to help express the idea. Thus bound variables dont represent any object of the statement.
- The truth set of a statement P(x) is the set of all values of x that make the statement P(x) true.
- The set of all possible values of variables is call
*universe of discourse*. Or variables*range*over this universe. - In general, means the same thing as .

### Solutions

**Soln1**

**(a)** where means is divisible by .

**(b)** where means is divisible by .

**(c)** where .

**Soln2**

**(a)** where is “x is men”, means “x is taller than y”.

**(b)** where means “x has brown eyes” and “x has brown hairs” respectively.

**(c)** where means “x has brown eyes” and “x has brown hairs” respectively.

**Soln3**

**(a)**

**(b)**

**(c)**

**(d)**

**Soln4**

**(a)**

**(b)**

**(c)**

**Soln5**

**(a)** . No free variables in the statement. Statement is true.

**(b)** . No free variables in the statement. Statement is false.

**(c)** . One free variable(c) in the statement. (Thanks Maxwell for the correction)

**Soln6**

**(a)** . There are two free variables and .

**(b)** . The statement has no free variables. It is a true statement.

**(c)** . The statement has no free variables. It is a false statement.

**Soln7**

**(a)** {Conrad Hilton Jr., Michael Wilding, Michael Todd, Eddie Fisher, Richard Burton, John Warner, Larry Fortensky}.

**(b)**

**(c)** { Daniel Velleman }

**Soln8**

**(a)** {1, 3}

**(b)**

**(c)**

Update:

As pointed out in comments, I got this wrong first time. Here is the correct answer:

or, equivalently .

Old Answer: