The logical statements that contain the basic connectives
and the negation are related to the statements containing
union, intersection, and complements as follows:
Let p be “The object belongs to set P.” Let q be
“The object belongs to set Q.”
p ^ q is equivalent to P n Q.
p ^ q is equivalent to P u Q.
p ^ q is equivalent to P u Q.
~p is equivalent to P.
In an argument, the premises and conclusion can be
translated into equivalent set statements, and Venn
diagrams can be drawn for the statements. The argument
is valid if the Venn diagram for the conclusion is a subset
of the intersection of the premises. Using Venn diagrams,
determine whether or not these arguments are valid.
(a) p -->q (b) p -->q
p ~p
=q =q
Electrical circuits are designed using truth tables. A
circuit consists of switches. Two switches wired in
series can be represented as p ^ q. Two switches wired
in parallel can be represented as p ^q.
In a series, circuit electricity will only flow when
both switches, p and q, are closed. In a parallel circuit,
electricity will flow when one or the other or both
switches are closed. In a truth table, T represents a
closed switch and F represents an open switch. Hence
the truth table for p ^ q shows electricity flowing only
when both switches are closed.
Truth table
p q p^ q
T T T
T F F
F T F
F F F
Also, when switch p is closed, switch ~p will be
open and vice versa, and p and ~p are different
switches. Using this knowledge, design a circuit for a
hall light that has switches at both ends of the hall such
that the light can be turned on or off from either switch.
Circuit
p q p ^ q
closed closed current
closed open no current
open closed no current
open open no current
and the negation are related to the statements containing
union, intersection, and complements as follows:
Let p be “The object belongs to set P.” Let q be
“The object belongs to set Q.”
p ^ q is equivalent to P n Q.
p ^ q is equivalent to P u Q.
p ^ q is equivalent to P u Q.
~p is equivalent to P.
In an argument, the premises and conclusion can be
translated into equivalent set statements, and Venn
diagrams can be drawn for the statements. The argument
is valid if the Venn diagram for the conclusion is a subset
of the intersection of the premises. Using Venn diagrams,
determine whether or not these arguments are valid.
(a) p -->q (b) p -->q
p ~p
=q =q
Electrical circuits are designed using truth tables. A
circuit consists of switches. Two switches wired in
series can be represented as p ^ q. Two switches wired
in parallel can be represented as p ^q.
In a series, circuit electricity will only flow when
both switches, p and q, are closed. In a parallel circuit,
electricity will flow when one or the other or both
switches are closed. In a truth table, T represents a
closed switch and F represents an open switch. Hence
the truth table for p ^ q shows electricity flowing only
when both switches are closed.
Truth table
p q p^ q
T T T
T F F
F T F
F F F
Also, when switch p is closed, switch ~p will be
open and vice versa, and p and ~p are different
switches. Using this knowledge, design a circuit for a
hall light that has switches at both ends of the hall such
that the light can be turned on or off from either switch.
Circuit
p q p ^ q
closed closed current
closed open no current
open closed no current
open open no current