SUNY Geneseo Department of Mathematics
Monday, February 11
Math 239 01
Spring 2019
Prof. Doug Baldwin
Problem set 2 in Canvas accidentally has solutions as well as questions. I’ll leave them there, but will look extra hard for evidence that you understand how to solve the problems and similar ones when I grade.
Parts of section 2.3
Give example(s) of members of the truth set, and non-members of the truth set, for the following predicates and universal sets:
Think of a predicate as a function whose body is a statement once its argument(s) is substituted for a variable. The truth set is then the set of values you can substitute for the variable in order to get a true statement. The universal set is where you can pick values for the variable (regardless of whether they make the statement true or false) from.
P(x) = “x is a prime number.” U = ℕ.
Some elements of the truth set: x = 3, x = 7
Some elements of the universal set not in the truth set: x = 4, x = 8
What if U = ℚ?
The truth set doesn’t change,
But the set of values that make the predicate false changes considerably: in addition to its previous values, it also contains numbers such as x = 3/4, x = 3 1/2.
Q(x) = “x > 7.” U = ℝ.
Some elements of the truth set: x = 7.0001, x = 8
Some elements of the universal set that aren’t in the truth set: x = 7, x = 6
What if U = ℤ?
Now the truth set contains, for example, x = 8, x = 9
And examples not in the truth set include: x = 6, x = 7
M(y) = “y > 7 and y2 = 100.” U = ℤ
This time the truth set can be written out in full: { 10 }
State the idea of “a number is the cube root of 1” as a predicate and its universal set.
P(x) = 3√1 = x, U = ℤ
P(x) = x3 = 1, U = C (the set of complex numbers)
The universal set makes a big difference for the truth set here. If U = ℤ, then the truth set is just {1}, but if U = C it’s { 1, -1/2 ± √3/2 i }
What predicates are, and related terminology of universal sets and truth sets.
The universal set is important to a predicate’s truth set.
Use set builder notation to define the interval [0,1) on the real number line.
Idea 1: { 0 ≤ x < 1 | x ∈ ℝ }. This is unfortunately backwards, you can’t have a predicate such as “0 ≤ x < 1” in the first part of a set builder expression.
Idea 2: maybe { [0,1) | x ∈ ℝ }. But the interval would need to be used in some sort of predicate, which should be in the second part of the expression.
Idea 3: { x ∈ ℝ | 0 ≤ x < 1 }. This works well — when using set builder notation to pick out those members of some large set (ℝ in this case) that are in the set you’re describing, the predicate that does the picking goes in the second part of the expression, and the first part uses the large set to define the predicate’s variable.
Idea 4: { x ∈ ℝ | x ∈ [0,1) } This also works, similarly to how Idea 3 worked.
More discussion of set builder notation.
Hopefully we’ll also start statements about all or some members of a set — quantifiers.
Start reading section 2.4 (though we’ll talk about it over two classes).