Question

Scenario:

Four people are working on a project. The people are numbered 1, 2, 3, and 4. The predicate C(x,y) indicates whether x has called y. So, C(2,3) is read, “Person 2 has called Person 3.” The table below shows the truth value of the predicate C(x,y) for each (x,y) pair. The truth value in row x and column y gives the truth value for C(x,y).

C (x,y)	1	2	3	4
1	F	T	F	T
2	T	F	T	T
3	F	F	F	T
4	F	F	F	F

A.  Translate each of the following quantified statements into English sentences:

1.  ∀x ∃y C(x,y)

2.  ∀y ∃x C(x,y)

3.  ∃y ∀x C(x,y)

4.  ∃x ∀y ((x ≠ y) → C(x,y))

5.  ∀x ∀y ((x ≠ y) → (C(x,y) ∨ C(y,x)))

B.  Indicate whether each of the following quantified statements is true or false. Justify your answers.

1.  ∀x ∃y C(x,y)

2.  ∀y ∃x C(x,y)

3.  ∃y ∀x C(x,y)

4.  ∃x ∀y ((x ≠ y) → C(x,y))

5.  ∀x ∀y ((x ≠ y) → (C(x,y) ∨ C(y,x)))

C.  Identify which statement from part A is equivalent to the following quantified statement: ¬ ∀y ∃x ¬ C(x,y). Prove your answer using de Morgan’s law, showing all steps.

Answer 1