Question

logic
use the following statements to write a compound statement for each conjunction or disjunction. then find its truth - value.
p: 60 seconds = 1 minute
q: congruent supplementary angles each have a measure of 90.
r: - 12 + 11 < - 1

1. p ∧ q
2. q ∨ r
3. ¬p ∨ q
4. ¬p ∧ ¬r

complete each truth table.
5.

p	q	¬p	¬q	¬p ∨ ¬q
t	t
t	f
f	t
f	f

6.

p	q	¬p	¬p ∨ q	p ∧ (¬p ∨ q)
t	t
t	f
f	t
f	f

construct a truth table for each compound statement.

1. q ∨ (p ∧ ¬q)
2. ¬q ∧ (¬p ∨ q)
3. school the venn diagram shows the number of students in the band who work after school or on the weekends.

a. how many students work after school and on weekends?
b. how many students work after school or on weekends?

Explanation:

Step1: Determine truth - values of p, q, r

$p$: 60 seconds = 1 minute is True (T). $q$: Congruent supplementary angles each have a measure of 90 is True (T). $r$: $- 12+11=-1$ and $-1=-1$ so $r$ is False (F).

Step2: Analyze $p\land q$

The $\land$ (and) operator is True only when both statements are True. Since $p = T$ and $q = T$, $p\land q$ is True.

Step3: Analyze $q\lor r$

The $\lor$ (or) operator is True when at least one statement is True. Since $q = T$ and $r = F$, $q\lor r$ is True.

Step4: Analyze $

eg p\lor q$
$
eg p$ is False since $p$ is True. Since $
eg p = F$ and $q = T$, $
eg p\lor q$ is True.

Step5: Analyze $

eg p\land
eg r$
$
eg p$ is False and $
eg r$ is True. The $\land$ operator requires both to be True, so $
eg p\land
eg r$ is False.

Step6: Complete truth - table for $

eg p\lor
eg q$
When $p = T,q = T$, $
eg p = F,
eg q = F,
eg p\lor
eg q=F$. When $p = T,q = F$, $
eg p = F,
eg q = T,
eg p\lor
eg q=T$. When $p = F,q = T$, $
eg p = T,
eg q = F,
eg p\lor
eg q=T$. When $p = F,q = F$, $
eg p = T,
eg q = T,
eg p\lor
eg q=T$.

Step7: Complete truth - table for $p\land(

eg p\lor q)$
When $p = T,q = T$, $
eg p = F,
eg p\lor q=T,p\land(
eg p\lor q)=T$. When $p = T,q = F$, $
eg p = F,
eg p\lor q=F,p\land(
eg p\lor q)=F$. When $p = F,q = T$, $
eg p = T,
eg p\lor q=T,p\land(
eg p\lor q)=F$. When $p = F,q = F$, $
eg p = T,
eg p\lor q=T,p\land(
eg p\lor q)=F$.

Step8: Analyze $q\lor(p\land

eg q)$
When $p = T,q = T$, $p\land
eg q = F,q\lor(p\land
eg q)=T$. When $p = T,q = F$, $p\land
eg q = T,q\lor(p\land
eg q)=T$. When $p = F,q = T$, $p\land
eg q = F,q\lor(p\land
eg q)=T$. When $p = F,q = F$, $p\land
eg q = F,q\lor(p\land
eg q)=F$.

Step9: Analyze $

eg q\land(
eg p\lor q)$
When $p = T,q = T$, $
eg q = F,
eg p = F,
eg p\lor q=T,
eg q\land(
eg p\lor q)=F$. When $p = T,q = F$, $
eg q = T,
eg p = F,
eg p\lor q=F,
eg q\land(
eg p\lor q)=F$. When $p = F,q = T$, $
eg q = F,
eg p = T,
eg p\lor q=T,
eg q\land(
eg p\lor q)=F$. When $p = F,q = F$, $
eg q = T,
eg p = T,
eg p\lor q=T,
eg q\land(
eg p\lor q)=T$.

Step10: Solve Venn - diagram part a

The number of students who work after school and on weekends is the intersection of the two sets. From the Venn - diagram, it is 3.

Step11: Solve Venn - diagram part b

The number of students who work after school or on weekends is $5 + 3+17=25$.

Answer:

1. True
2. True
3. True
4. False

5.

$p$

$q$

$

eg p$ | $
eg q$ | $
eg p\lor
eg q$ |

T	T	F	F	F
T	F	F	T	T
F	T	T	F	T
F	F	T	T	T

6.

$p$

$q$

$

eg p$ | $
eg p\lor q$ | $p\land(
eg p\lor q)$ |

T	T	F	T	T
T	F	F	F	F
F	T	T	T	F
F	F	T	T	F

7.

$p$

$q$

$p\land

eg q$ | $q\lor(p\land
eg q)$ |

T	T	F	T
T	F	T	T
F	T	F	T
F	F	F	F

8.

$p$

$q$

$

eg q$ | $
eg p$ | $
eg p\lor q$ | $
eg q\land(
eg p\lor q)$ |

T	T	F	F	T	F
T	F	T	F	F	F
F	T	F	T	T	F
F	F	T	T	T	T

1. a. 3

b. 25