Discrete mathematics (chapter 1)
study guide correctly solved 2026
proposition - correct answer ✔A proposition is a declarative sentence that is either true or
false.
Conjunction - correct answer ✔p∧q= "p and q" = proposition that is true when
both p and q are true, and is false otherwise.
Disjunction - correct answer ✔p∨q= "p or q" = proposition that is false when both
p and q are false, and is true otherwise.
Exclusive or - correct answer ✔p ⊕q = "only p or only q" = proposition that is true
when exactly one of p and q is true and is false otherwise.
Conditional statement (implication) - correct answer ✔p →q = proposition that is
false when p is true and q is false, and is true otherwise.
Biconditional statement (bi-implications) - correct answer ✔p ↔q = proposition that is true when p
and q have the same truth values, and is false otherwise.
Express the statement p→q as a statement in English - correct answer ✔"if p, then q"
"p implies q"
"if p,q" "p only if q"
"p is sufficient for q"
, "a sufficient condition for q is p"
"q if p"
"q whenever p"
"q when p"
"q is necessary for p"
"a necessary condition for p is q"
"q follows from p"
"q unless ¬p"
p only if q - correct answer ✔To remember that "p only if q" expresses the same thing as "if p, then q,"
*Note that "p only if q" says that p cannot be true when q is not true. => ¬q→¬p
p: "you are a member of the team"
q: "you take afternoon classes"
p→¬q : "your are a member of the team only if you don't take afternoon classes"
q unless ¬p - correct answer ✔expresses the same conditional statement as "if p, then q,"
note that "q unless ¬p" means that if ¬p is false, then q must be true. That is, the statement "q unless
¬p" is false when p is true but q is false, but it is true otherwise. Consequently,"q unless ¬p" and p→q
always have the same truth value.=> p→q
"p necessary q" - correct answer ✔?????
How can this English sentence be translated into a logical expression?
"You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old." -
correct answer ✔Solution:Let q,r, and s represent "You can ride the roller coaster," "You are under 4
feet tall," and "You are older than 16 years old," respectively.
study guide correctly solved 2026
proposition - correct answer ✔A proposition is a declarative sentence that is either true or
false.
Conjunction - correct answer ✔p∧q= "p and q" = proposition that is true when
both p and q are true, and is false otherwise.
Disjunction - correct answer ✔p∨q= "p or q" = proposition that is false when both
p and q are false, and is true otherwise.
Exclusive or - correct answer ✔p ⊕q = "only p or only q" = proposition that is true
when exactly one of p and q is true and is false otherwise.
Conditional statement (implication) - correct answer ✔p →q = proposition that is
false when p is true and q is false, and is true otherwise.
Biconditional statement (bi-implications) - correct answer ✔p ↔q = proposition that is true when p
and q have the same truth values, and is false otherwise.
Express the statement p→q as a statement in English - correct answer ✔"if p, then q"
"p implies q"
"if p,q" "p only if q"
"p is sufficient for q"
, "a sufficient condition for q is p"
"q if p"
"q whenever p"
"q when p"
"q is necessary for p"
"a necessary condition for p is q"
"q follows from p"
"q unless ¬p"
p only if q - correct answer ✔To remember that "p only if q" expresses the same thing as "if p, then q,"
*Note that "p only if q" says that p cannot be true when q is not true. => ¬q→¬p
p: "you are a member of the team"
q: "you take afternoon classes"
p→¬q : "your are a member of the team only if you don't take afternoon classes"
q unless ¬p - correct answer ✔expresses the same conditional statement as "if p, then q,"
note that "q unless ¬p" means that if ¬p is false, then q must be true. That is, the statement "q unless
¬p" is false when p is true but q is false, but it is true otherwise. Consequently,"q unless ¬p" and p→q
always have the same truth value.=> p→q
"p necessary q" - correct answer ✔?????
How can this English sentence be translated into a logical expression?
"You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old." -
correct answer ✔Solution:Let q,r, and s represent "You can ride the roller coaster," "You are under 4
feet tall," and "You are older than 16 years old," respectively.
