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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.

Quantifiers and connectives 2

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.

Questions and exercises

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?

Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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