WGU C960 DISCRETE MATH II OA
COMPREHENSIVE STUDY GUIDE WITH
SOLUTIONS
◉ 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.
◉ direct proofs
Answer: used for conditional statements
If p then q
Assume p
Therefore q
◉ proofs by contrapositive
COMPREHENSIVE STUDY GUIDE WITH
SOLUTIONS
◉ 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.
◉ direct proofs
Answer: used for conditional statements
If p then q
Assume p
Therefore q
◉ proofs by contrapositive
