discrete math 1 wgu Exam Questions And Answers
Exclusive or. ⊕ - ANS One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: - ANS disjunction
Order of operations in absence of parentheses. - ANS 1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is th...
Exclusive or. ⊕ - ANS One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: - ANS disjunction
K
Order of operations in absence of parentheses. - ANS 1. ¬ (not)
2. ∧ (and)
C
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
LO
truth table with three variables - ANS see pic
2^3 rows
proposition - ANS p → q
Ex: If it is raining today, the game will be canceled.
YC
Converse: - ANS q → p
If the game is canceled, it will rain today.
Contrapositive - ANS ¬q → ¬p
D
If the game is not cancelled, then it is not raining today.
U
Inverse: - ANS ¬p → ¬q
If it is not raining today, the game will not be cancelled.
ST
biconditional - ANS p ↔ q
true when P and Q have the same truth value.
see truth table pic.
free variable - ANS ex.
P(x)
the variable is free to take any value in the domain
bound variable - ANS ∀x P(x)
,bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x), - ANS 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. - ANS ¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
using a truth table to establish the validity of an argument - ANS see pic.
K
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.
C
A valid argument is a guarantee that the conclusion is true whenever all of the hypotheses are true.
LO
If when the hypotheses are true, the conclusion is not, then it is invalid.
YC
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. - ANS see pic.
D
theorem - ANS any statement that you can prove
U
proof - ANS 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.
ST
the proof of a theorem may make use of axioms: - ANS which are statements assumed to be true.
proofs by exhaustion - ANS trying everything in the given universe.
proofs by counter example - ANS 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 - ANS used for conditional statements
If p then q
Assume p
Therefore q
proofs by contrapositive - ANS proves a conditional theorem of the form p → q by showing that the
contrapositive ¬q → ¬p is true. In other words, ¬c is assumed to be true and ¬p is proven as a result of
¬q.
Logically equivalent to if p then q
K
proof by contradiction - ANS (indirect proof)
starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a
C
result of this assumption.
Notice not a conditional.
Want to prove Y
LO
Assume not Y
Find a contradiction in X and Not Y
Therefore, claim not not Y.
proof by cases - ANS A proof by cases of a universal statement such as ∀x P(x) breaks the domain for
YC
the variable x into different classes and gives a different proof for each class. Every value in the domain
must be included in at least one class.
Unit 2 sets and functions - ANS
object in a set are called - ANS elements
D
The symbol ∈ is used: - ANS to indicate that an element is in a set, as in 2 ∈ A
U
The set with no elements is called the empty set and is denoted by the symbol ∅. - ANS The empty set is
sometimes referred to as the null setand can also be denoted by {}. Because the empty set has no
elements, for any element a, a ∉ ∅ is true.
ST
the cardinality of the empty set is: - ANS zero
N - ANS The set of natural numbers: All integers greater than or equal to 0. 0, 1, 2, ...
Z - ANS The set of all integers. ..., -2, -1, 0, 1, 2, ...
Q - ANS The set of rational numbers: All real numbers that can be expressed as a/b, where a and b are
integers and b ≠ 0. 0, 1/2, 5.23, -5/3
R - ANS The set of real numbers.
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