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: