IB Math Studies 1 Logic Test 3 Review ANSWERS
CATEGORIES:
Converse, Inverse, Contrapostive
10) State the inverse: If I do my homework, then I have an A on the test. (negate both)

If I do NOT do my homework, then I do NOT get an A on the test.
20) State the contrapositive: If it is snowing, then we will not have school. (flip, negate both)

If we have school, then it is not snowing.
30) Given….. p: I like cheese q: I like pizza. Write the following in symbols:
If I do not like pizza, then I do not like cheese.

~q ⇒ ~p
40) Prove that the converse and inverse are logically equivalent.
Must prove using a truth table!
p
q
~p
~q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
q ⇒p
(converse)
T
T
F
T
~p ⇒ ~q
(inverse)
T
T
F
T
50) Prove that the implication and the contrapositive are logically equivalent.
p
q
~p
~q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
p ⇒q
(implication)
T
F
T
T
~q ⇒ ~p
(contrapositive)
T
F
T
T
Truth Tables
10) Create a truth table for ~ p Λ ~ q
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
20) Create a truth table for
p
T
T
F
F
q
T
F
T
F
~q
F
T
F
T
~p Λ ~q
F
F
F
T
(p V q) Λ q
pVq
T
T
T
F
(p V q) Λ q
T
F
T
F
30) Create a truth table for ~ (p V q) Λ q
p
T
T
F
F
q
T
F
T
F
pVq
F
T
T
F
~( p V q )
T
F
F
T
~ (p V q) Λ q
T
F
F
F
40) Create a truth table for ~ (p V q) V ~ q
p
T
T
F
F
q
T
F
T
F
~q
F
T
F
T
pVq
F
T
T
F
~( p V q )
T
F
F
T
50) Determine if the following are logically equivalent:
p
T
T
F
F
~p
F
F
T
T
q
T
F
T
F
~q
F
T
F
T
~ (p V q) V q
T
T
F
T
~(p Λ q) = ~p V ~q
pΛq
T
F
F
F
~( p Λ q )
F
T
T
T
~ p V ~q
F
T
T
T
Truth Tables Implications
10) Construct a truth table for the following
p
T
T
F
F
q
T
F
T
F
p↔q
T
F
F
T
~ (p ↔ q)
~ (p ↔ q)
F
T
T
F
20) Determine if the following are tautologies, logical contradictions, or neither. ( p V q) → ( ~p)
p
T
T
F
F
~p
F
F
T
T
q
T
F
T
F
pVq
T
T
T
F
( p V q) → ( ~p)
F
F
T
T
30) Determine if the following are tautologies, logical contradictions, or neither. (p Λ q) → (p V q)
p
T
T
F
F
q
T
F
T
F
pΛq
T
F
F
F
pVq
T
T
T
F
(p Λ q) → (p V q)
T
T
T
T
40) Determine if the following are tautologies, logical contradictions, or neither. p Λ (p ↔ q)
p
T
T
F
F
q
T
F
T
F
p↔q
T
F
F
T
p Λ (p ↔ q)
T
F
F
F
50) Determine if the following are tautologies, logical contradictions, or neither.
(p → ~q) V (~p → q)
p
~p
q
~q
p → ~q
~p → q
T
T
F
F
F
F
T
T
T
F
T
F
F
T
F
T
F
T
T
T
T
T
T
F
(p → ~q) V (~p → q)
T
T
T
T
Truth Sets and Valid Arguments
10)List the truth sets for U, P, and P’ : U = { x⃓ 0 < x ≤ 18, x є N}, p: the set of prime numbers



U = { 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
P = {2,3,5,7,11,13,17}
P’ = {1,4,6,8,9,10, 12,14,15,16, 18}
--UNIVERSAL SET
--PRIME NUMBERS B/W 1 and 18
--EVERYTHING NOT IN P
20) List the truth sets and draw a diagram
U = { x⃓ 0 < x ≤ 20, x є N}



p: even numbers
q: prime numbers
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
P = {2,4,6,8,10,12,14,16,18,20}
Q = {2,3,5,7,11,13,17,19}
4,6,8,10,12,
14,16,18,20
30) Determine the validity of the argument
3,5,7,11,
13,17,19
40) Determine the validity of the argument
50) Determine the validity of the argument NOT VALID (Valid means ALL true)
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
pΛq
T
T
F
F
F
F
F
F
(p Λ q) → r
T
F
T
T
T
T
T
T
(p Λ q) → r Λ p
T
F
T
T
F
F
F
F
(p Λ q) → r Λ p → r
T
T
T
F
T
T
T
T
Venn Diagrams
10) Represent the following on a Venn Diagram: p Λ q
20) Represent the following on a Venn Diagram: p V q
30) Express in terms of P and Q

PVQ
40) Express in terms of P and Q

~P Λ Q
50) Represent using a Venn Diagram: q V (p Λ r)