Quiz. Term evaluation (FOL) [0014]

  • November 22, 2025

Consider the following domain/universe of discourse, variable assignment, and interpretation:

  • Universe \[ U \triangleq \{1, 2\} \]
  • Variable assignment \[ \sigma \triangleq \{x\mapsto 2, y\mapsto 1\} \]
  • Interpretation \[ \begin {align*} I & \triangleq \{a \mapsto 1, b \mapsto 2\} \\ I & \triangleq \{f(1, 1) \mapsto 2, f(1, 2) \mapsto 2, f(2, 1) \mapsto 1, f(2, 2) \mapsto 1\} \end {align*} \]

What is the evaluation of the following terms under the given interpretation and variable assignment?

  1. \(f(a, y)\)
  2. \(f(x, b)\)
  3. \(f(f(x, b), f(a, y))\)

Solution. 

  • November 22, 2025
  1. \(\langle I, \sigma \rangle (f(a, y)) = I(f)(\langle I, \sigma \rangle (a), \langle I, \sigma \rangle (y)) = I(f)(I(a), \sigma (y)) = I(f)(1, 1) = 2\)
  2. \(\langle I, \sigma \rangle (f(x, b)) = I(f)(\langle I, \sigma \rangle (x), \langle I, \sigma \rangle (b)) = I(f)(\sigma (x), I(b)) = I(f)(2, 2) = 1\)
  3. \(\langle I, \sigma \rangle (f(f(x, b), f(a, y))) = I(f)(\langle I, \sigma \rangle (f(x, b)), \langle I, \sigma \rangle (f(a, y))) = I(f)(1, 2) = 2\)