Elementary Set Theory Problems

Elementary Set Theory Problems

Simplify the following statements. Which variables are free and which are
bound? If the statement has no free variables, say whether it is true or
false.
(a) w ∈ {x ∈ R | 13 − 2x > c}.
(b) 4 ∈ {x ∈ R | 13 − 2x ∈ {y | y is a prime
number}}. (It might make this statement easier to read if we let P = {y |
y is a prime number}; using this notation, we could rewrite the statement
as 4 ∈ {x ∈ R | 13 − 2x ∈ P}.)
(c) 4 ∈ {x ∈{y | y is a prime number} |13 − 2x > 1}. ,
{y | y is a prime number} = P
Solutions:
(a) (w ∈ R) ∧ (13 − 2(4) > c) x us bound, c,w are free
variablses
(b) (4 ∈ R) ∧ (13 − 2(4) ∈ P) bound variables x,
y, no free variables TRUE
(4 ∈ R) ∧ (4 ¬∈ P) ∧ (13 − 2(4) > 1) bound
variables x, y, no free varaibles FALSE