Recipe
Find the truth set of a predicate
Use when: you're given a predicate P(x) and a finite, listable domain, and asked which elements make P(x) true
Goal: the complete truth set, the subset of the domain where the predicate actually holds
Taught in Predicates and truth sets
Quick recipe
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1
Write out the domain as a finite list. A truth set can only be found by checking elements one at a time, so nothing can be skipped.
Example
P(x): "x is even," over the domain \{1,2,3,4,5,6\}. The domain to check is 1, 2, 3, 4, 5, 6.
Why: skipping an element risks leaving it out of the truth set, or wrongly including it, without ever actually checking.
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2
Plug each domain element into the predicate in turn, then simplify the resulting statement until its truth value is obvious.
Example
P(1): 1 is even, false. P(2): 2 is even, true. P(3): 3 is even, false. P(4): 4 is even, true. P(5): 5 is even, false. P(6): 6 is even, true.
Why: a predicate has no truth value of its own until a specific value is substituted in for the variable.
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3
Gather every domain element whose evaluation was true, and drop the rest. This collected list is the truth set.
Example
P(x) was true at 2, 4, and 6, so the truth set is \{2, 4, 6\}.
Why: the truth set is defined as exactly the domain elements where the predicate holds, nothing more and nothing less.
Common mistakes
- Testing only some domain elements instead of every single one.
- Writing P(2) into the answer instead of the element 2 itself -- the truth set holds elements, not evaluated predicates.
- Rushing the substitution step and making an arithmetic or comparison slip that flips a true case to false or back.
- Assuming ahead of time that the truth set must be nonempty, or must equal the whole domain.