WGU D420 DISCRETE MATH 1 EXAM 2026
QUESTIONS AND ANSWERS GUARANTEED
PASS
⩥ inclusive or is a:. Answer: disjunction
⩥ Order of operations in absence of parentheses.. Answer: 1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then
disjunction:
⩥ truth table with three variables. Answer: see pic
2^3 rows
⩥ proposition. Answer: p → q
Ex: If it is raining today, the game will be cancelled.
⩥ Converse:. Answer: q → p
If the game is cancelled, it is raining today.
,⩥ Contrapositive. Answer: ¬q → ¬p
If the game is not cancelled, then it is not raining today.
⩥ Inverse:. Answer: ¬p → ¬q
If it is not raining today, the game will not be cancelled.
⩥ biconditional. Answer: p ↔ q
true when P and Q have the same truth value.
see truth table pic.
⩥ free variable. Answer: ex.
P(x)
the variable is free to take any value in the domain
⩥ bound variable. Answer: ∀x P(x)
bound to a quantifier.
⩥ In the statement (∀x P(x)) ∧ Q(x),. 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.. Answer:
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
⩥ using a truth table to establish the validity of an argument. 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.. Answer: see pic.
⩥ theorem. Answer: any statement that you can prove
⩥ proof. 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:. Answer: which are
statements assumed to be true.
⩥ proofs by exhaustion. Answer: trying everything in the given universe.
⩥ proofs by counter example. Answer: show that one fails.
A counterexample is an assignment of values to variables that shows that
a universal statement is false.
A counterexample for a conditional statement must satisfy all the
hypotheses and contradict the conclusion.
QUESTIONS AND ANSWERS GUARANTEED
PASS
⩥ inclusive or is a:. Answer: disjunction
⩥ Order of operations in absence of parentheses.. Answer: 1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then
disjunction:
⩥ truth table with three variables. Answer: see pic
2^3 rows
⩥ proposition. Answer: p → q
Ex: If it is raining today, the game will be cancelled.
⩥ Converse:. Answer: q → p
If the game is cancelled, it is raining today.
,⩥ Contrapositive. Answer: ¬q → ¬p
If the game is not cancelled, then it is not raining today.
⩥ Inverse:. Answer: ¬p → ¬q
If it is not raining today, the game will not be cancelled.
⩥ biconditional. Answer: p ↔ q
true when P and Q have the same truth value.
see truth table pic.
⩥ free variable. Answer: ex.
P(x)
the variable is free to take any value in the domain
⩥ bound variable. Answer: ∀x P(x)
bound to a quantifier.
⩥ In the statement (∀x P(x)) ∧ Q(x),. 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.. Answer:
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
⩥ using a truth table to establish the validity of an argument. 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.. Answer: see pic.
⩥ theorem. Answer: any statement that you can prove
⩥ proof. 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:. Answer: which are
statements assumed to be true.
⩥ proofs by exhaustion. Answer: trying everything in the given universe.
⩥ proofs by counter example. Answer: show that one fails.
A counterexample is an assignment of values to variables that shows that
a universal statement is false.
A counterexample for a conditional statement must satisfy all the
hypotheses and contradict the conclusion.
