discrete math 1 wgu questions and
answers
Exclusive or. ⊕ - answer One or the other, but not both.
We can go to the park or the movies.
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.
direct proofs - answer used for conditional statements
answers
Exclusive or. ⊕ - answer One or the other, but not both.
We can go to the park or the movies.
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.
direct proofs - answer used for conditional statements
