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3. ∃x [ P(x) ⋁Q(x) ] ⇔[ ∃x P(x) ⋁∃x Q(x) ], again for the same example, can be shown in Figure 3:
LHS says someone is rich or happy, and RHS says someone is rich or someone is happy. Thus clearly LHS implies RHS. Also if someone is rich then that person is certainly rich or happy. Thus RHS implies LHS.
4. ∃x [ P(x) ⋀Q(x) ] ⇒[ ∃x P(x) ⋀∃x Q(x) ], for the same example, can be shown in Figure 4:
LHS say someone is rich and happy. Hence there is someone who is rich and there is someone who is happy. Hence LHS implies RHS. However, since RHS can be true without anyone being rich and happy at the same time, RHS does not necessarily imply LHS.
If a wff (Q below) in the scope of a quantifier does not have the variable (x below) that is quantified by that quantifier, then that wff can be taken out of the scope of that quantifier. That is,
1. ∀x [ P(x) ⋀Q ] ⇔[ ∀x P(x) ⋀Q ]
2. [ ∀x P(x) ⋁Q ] ⇔∀x [ P(x) ⋁Q ]
3. ∃x [ P(x) ⋁Q ] ⇔[ ∃x P(x) ⋁Q ]
4. ∃x [ P(x) ⋀Q ] ⇔[ ∃x P(x) ⋀Q ],
where Q in all these formulas DO NOT have the variable x .
Note: When implication → and/or equivalence ↔ are involved, you can not necessarily take Q outside the scope. To see what happens, express → and ↔ using ⋁and ⋀, and apply the above formulas. For example ∀x [ P(x)→Q ] is NOT equivalent to ∀x P(x)→Q . Rather it is equivalent to ∃x P(x)→Q. Further details are left as an exercise.
1. Which of the following sentences is a proposition?
a. Every one is happy.
b. If it snows, then schools are closed in Norfolk, VA.
c. x + 2 is positive
d. Take an umbrella with you.
e. I suggest that you take an umbrella with you
2. Which of the following tables is a truth table?
Z below represents a proposition involving P and Q.
| Table 1 | ||
| P | Q | Proposition Z |
| F | F | F |
| T | F | T |
| T | T | T |
| T | F | T |
| Table 2 | ||
| P | Q | Proposition Z |
| F | F | F |
| T | F | T |
| T | T | F |
| Table 3 | ||
| P | Q | Proposition Z |
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
| Table 4 | |
| P | Proposition Z |
| F | F |
| F | T |
| T | F |
3. Indicate which of the following statements are correct and which ones are incorrect.
a. If P is True and Q is False, then P⋀Q is True.
b. If P is False and Q is True, then P → Q is True.
c. If P is False and Q is False, then P ↔ Q is False
d. If P is True and Q is False, then P ⋁ Q is True.
e. If P is True and Q is False, then ¬[P⋀Q] is False
4. Indicate which of the following expressions are propositions and which are not.
a. P⋀¬Q.
b. [[P ⋁ Q] → [Q ⋀ R]]
c. [¬[P ↔ ⋀ Q ] ⋁ Q ]
d. [¬¬P ⋁ Q]
e. [[Q ⋁ R][P ⋀ Q]]
5. Indicate which of the following converses and contrapositives are correct and which are not.
a. If it snows, the schools will be closed.
Converse: If the schools are closed, it snows.
Contrapositive: If the schools are not closed, it does not snow.
b. If I work all night, I can finish this project.
Converse: If I cannot finish this project, I work all night.
Contrapositive: If I can finish this project, I don’t work all night.
c. I eat spicy food, only if it upsets my stomach.
Converse: If I eat spicy food, it upsets my stomach.
Contrapositive: If I don’t eat spicy food, it doesn’t upset my stomach.
6. Which of the following pairs of propositions are logically equivalent?
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