Discrete math 1 wgu Question with
complete solution 2026
Exclusive or. ⊕ - correct answer ✔One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: - correct answer ✔disjunction
Order of operations in absence of parentheses. - correct answer ✔1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables - correct answer ✔see pic
2^3 rows
proposition - correct answer ✔p → q
Ex: If it is raining today, the game will be cancelled.
Converse: - correct answer ✔q → p
If the game is cancelled, it is raining today.
Contrapositive - correct answer ✔¬q → ¬p
,If the game is not cancelled, then it is not raining today.
Inverse: - correct answer ✔¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional - correct answer ✔p ↔ q
true when P and Q have the same truth value.
see truth table pic.
free variable - correct answer ✔ex.
P(x)
the variable is free to take any value in the domain
bound variable - correct answer ✔∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x), - correct answer ✔the variable x in P(x) is bound
the variable x in Q(x) is free.
this statement is not a proposition cause of the free variable.
summary of De Morgan's laws for quantified statements. - correct answer ✔¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
, using a truth table to establish the validity of an argument - correct answer ✔see pic.
In order to use a truth table to establish the validity of an argument, a truth table is constructed for all
the hypotheses and the conclusion.
A valid argument is a guarantee that the conclusion is true whenever all of the hypotheses are true.
If when the hypotheses are true, the conclusion is not, then it is invalid.
the argument works if every time the hypotheses (anything above the line) are true, the conclusion is
also true.
hypotheses dont always all need to be true, see example. but every time all the hypotheses are true, the
conclusion needs to be true as well.
rules of inference. - correct answer ✔see pic.
theorem - correct answer ✔any statement that you can prove
proof - correct answer ✔A proof consists of a series of steps, each of which follows logically from
assumptions, or from previously proven statements, whose final step should result in the statement of
the theorem being proven.
the proof of a theorem may make use of axioms: - correct answer ✔which are statements assumed to
be true.
complete solution 2026
Exclusive or. ⊕ - correct answer ✔One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: - correct answer ✔disjunction
Order of operations in absence of parentheses. - correct answer ✔1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables - correct answer ✔see pic
2^3 rows
proposition - correct answer ✔p → q
Ex: If it is raining today, the game will be cancelled.
Converse: - correct answer ✔q → p
If the game is cancelled, it is raining today.
Contrapositive - correct answer ✔¬q → ¬p
,If the game is not cancelled, then it is not raining today.
Inverse: - correct answer ✔¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional - correct answer ✔p ↔ q
true when P and Q have the same truth value.
see truth table pic.
free variable - correct answer ✔ex.
P(x)
the variable is free to take any value in the domain
bound variable - correct answer ✔∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x), - correct answer ✔the variable x in P(x) is bound
the variable x in Q(x) is free.
this statement is not a proposition cause of the free variable.
summary of De Morgan's laws for quantified statements. - correct answer ✔¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
, using a truth table to establish the validity of an argument - correct answer ✔see pic.
In order to use a truth table to establish the validity of an argument, a truth table is constructed for all
the hypotheses and the conclusion.
A valid argument is a guarantee that the conclusion is true whenever all of the hypotheses are true.
If when the hypotheses are true, the conclusion is not, then it is invalid.
the argument works if every time the hypotheses (anything above the line) are true, the conclusion is
also true.
hypotheses dont always all need to be true, see example. but every time all the hypotheses are true, the
conclusion needs to be true as well.
rules of inference. - correct answer ✔see pic.
theorem - correct answer ✔any statement that you can prove
proof - correct answer ✔A proof consists of a series of steps, each of which follows logically from
assumptions, or from previously proven statements, whose final step should result in the statement of
the theorem being proven.
the proof of a theorem may make use of axioms: - correct answer ✔which are statements assumed to
be true.
