WGU D420 Discrete Math 1 Exam Questions
with Verified Correct Answers
Exclusive or. ⊕
One or the other, but not both.
We can go to the park or the movies.
inclusive or is a:
disjunction
Order of operations in absence of parentheses.
1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables
see pic
2^3 rows
proposition
p→q
Ex: If it is raining today, the game will be cancelled.
Converse:
q→p
If the game is cancelled, it is raining today.
,Contrapositive
¬q → ¬p
If the game is not cancelled, then it is not raining today.
Inverse:
¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional
p↔q
true when P and Q have the same truth value.
see truth table pic.
free variable
ex.
P(x)
the variable is free to take any value in the domain
bound variable
∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x),
,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.
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
using a truth table to establish the validity of an argument
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.
see pic.
theorem
any statement that you can prove
proof
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:
which are statements assumed to be true.
proofs by exhaustion
trying everything in the given universe.
proofs by counter example
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.
direct proofs
used for conditional statements
with Verified Correct Answers
Exclusive or. ⊕
One or the other, but not both.
We can go to the park or the movies.
inclusive or is a:
disjunction
Order of operations in absence of parentheses.
1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables
see pic
2^3 rows
proposition
p→q
Ex: If it is raining today, the game will be cancelled.
Converse:
q→p
If the game is cancelled, it is raining today.
,Contrapositive
¬q → ¬p
If the game is not cancelled, then it is not raining today.
Inverse:
¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional
p↔q
true when P and Q have the same truth value.
see truth table pic.
free variable
ex.
P(x)
the variable is free to take any value in the domain
bound variable
∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x),
,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.
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
using a truth table to establish the validity of an argument
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.
see pic.
theorem
any statement that you can prove
proof
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:
which are statements assumed to be true.
proofs by exhaustion
trying everything in the given universe.
proofs by counter example
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.
direct proofs
used for conditional statements
