Which of the following is false? Read ∧ as AND, ∨ as OR, ∼ as NOT, → as one…
1996
Which of the following is false? Read ∧ as AND, ∨ as OR, ∼ as NOT, → as one way implication and ↔ as two way implication.
- A.
((x→y) ∧ x)→ y
- B.
((∼x→y) ∧ (∼x→∼y))→ x
- C.
(x→ (x ∨ y))
- D.
((x ∨ y) ↔ (∼x→∼y))
Attempted by 15 students.
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Correct answer: D
To find the false statement, let's look closely at Option D: ((x v y) <-> (~x -> ~y)).
For a two-way implication (equivalence, <->) to be true, the expressions on both the left and right sides must have identical truth values in every scenario. Let's test what happens when x = False and y = True:
Evaluate Left Side: (x v y) False v True = True
Evaluate Right Side: (~x -> ~y)
~x becomes ~False = True
~y becomes ~True = False
The implication becomes True -> False, which evaluates to False
Evaluate the full equivalence: Left Side <-> Right Side True <-> False = False
Because the statement evaluates to False in this scenario, it is a contingency, not a tautology (it is not universally true). Therefore, Option D is the false statement.
Note on the math: The expression ~x -> ~y is the inverse of the implication x -> y. It is not logically equivalent to x v y. The correct equivalence for x v y in implication form would be ~x -> y (the contrapositive of ~y -> x).