Propositions, compound statements and truth values
Description of the concept...
Learn about it
The following is a series of slides and videos that will help you understand, learn about and review this sub-topic.
Keep track of your progress, look at the revision cards and practice the exam questions on this Logic document
Slides Gallery
Use these slides to review the material and key points covered in the videos.
Logical Propositions
This is an introduction to the idea of symbolic logic, the language, the notation and the idea of compound logical statements.
Language, words and symbols
How do we take compound statements written in words and write them in symbols? What about the other way around? This video answers the questions.
Standard results - AND, OR, NEGATION
The formula booklet has a set of standard results for AND, OR and NEGATION. Here we explain those and get to the point that we know these off by heart because they make sense!
Standard Results - Implication
Implication has caused many a headache over the years and it is an excellent thought exercise to discuss it. This video explains how following the 'TRUE until proven FALSE' approach can bypass the headache! The first video covers straight implication, the second looks at If and only IF.
Constructing truth tables - 2 and 3 propositions
In these two videos we go over constructing and completing truth tables involving both 2 and 3 propositions. One column at a time, one row at a time, we are only ever comparing 2 truth values to deduce a third.
Contradiction, Tautology and Validity
What if all the truth values are 'False'? What if they are 'True'? These videos cover these important ideas with some examples leading up to the notion of a 'Valid Argument'
Logical Equivalence
What does it mean when to compound logical statements are said to be logically equivalent? Here is an explanation with a couple of examples to show the point.
Converse, Inverse and Contrapositive
These definitions are not given in your formula booklets and you need to have them committed to memory. This video explains what they are and what they mean.
Revise
This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.
Revision Cards
Review these condensed 'key point' Revision cards to help you check and keep ideas fresh in your mind.
Practice Questions
Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.
1
Q1 Words and symbols conversion in abstract
Consider the propositions p and q. Match the following statements with the correct logic notation for the statement.
A - Not p
B - p and q
C - p or q
D - p or q but not both
E - p and not q
F - If p THEN q
G - IF and only IF q THEN p
H - IF p and q THEN q
Enter the corresponding letter A to H in the box next to the expression
\(p\underset { - }{ \vee } q\)
\(p\wedge q\)
\(q\Leftrightarrow q\)
\(\neg p\)
\(p\wedge q\Rightarrow q\)
\(p\Rightarrow q\)
\(p\vee q\)
\(p\wedge \neg q\)
This question is about understanding the basics of the ntation. See the slide below.
2
Q2 3 props and compound statements to symbols
Consider the following propositions
p - Simba is lost and q - Simba is a child r - Simba is alone
Match the following compound logical statements with the correct symbols below.
A - Simba is lost and Simba is a child
B - Simba is lost or Simba is alone
C - Simba is not lost and Simba is not alone
D - IF Simba is lost THEN he is alone
E - Simba is child AND (Simba is either lost OR alone but not both)
F - (Simba is not alone and not a child) OR he is lost
G - IF Simba is a child OR he is lost THEN he is alone
H - IF Simba is not a child AND is alone THEN he is not lost
Again, see the slides about notation to help with this.
3
Q3 Standard Truth values, match the column to the notation
The table below shows some of the standard truth values for compound statements involving the 4 combinations of propositions p and q. Match the letters A - F with the compound statement that its truth value represent.
p
q
A
B
C
D
E
F
T
T
F
T
T
F
T
T
T
F
F
F
T
T
F
F
F
T
T
F
T
T
T
F
F
F
T
F
F
F
T
T
\(p\Rightarrow q\)
\(p\vee q\)
\(p\wedge q\)
\(p\Leftrightarrow q\)
\(p\underset { - }{ \vee } q\)
\(\neg p\)
This question is about being aware of the standard results. there is no need to memorise them as they are given in the formula booklet, BUT it does help to actually understand them....
4
Q4 Simple TTs table showing a, b, c, de, e etc…. Students enter T or F Implication, backwards and forwards
The Truth tables below have a number of blanks. Consider each blank carefully - in each case you are only require to combine the values in two other columns with one of the standrd results. THen enter either 'T' of 'F' in the box next to the lable for each blank.
Enter either 'T' or 'F' in each of the following.
a = b = c = d =
e = f = g = h =
i = j = k = l =
m = n =
a = F - should be F because p is T so not p must be the opposite.
b = F - should be F because q is T so not q must be the opposite
c = T - p is T and not q is T, this is an AND statement so they both need to be true and they are!
d = F - This is the same AND statement as above and we have an F and a T where we need both so F
e = T - This is an OR statement. Only one needs to be true and q is true so T
f = T - This is the same OR statement. In this case both are True and we only need one so T
g = F - An Exclusive OR statement so we only need one to be true (not both) . Neither are so this is F
h = T - In this case p is false and not q is true so we have T.
i = T - This is an implication. The only time we get an F is when the 1st is T and the 2nd F all others are T
j = T - For the same reason. Here we have F T which gives T
k = T - Here we have not p F then q F, for implication F F gives T
l = F - In this case we have not p T then q F, TF (in that order) is the only combination that leads to F
m = T - FF in that order gives us T
n = T - FT in that order gives us T
5
Q5 2 Propositions in context
Consider the propositions p - Zazu can fly, q - Zazu is bird
a) Which of the following is the correct notation for the compound statement 'If Zazu can not fly then he is not a bird.
A \(\neg (p\Rightarrow q)\) B \(\neg p\Rightarrow q\) C \(\neg p\Rightarrow \neg q\) D \(\neg q\Rightarrow \neg p\)
b) The following is a truth table to show truth values for the statement \(\neg p\Rightarrow \neg q\) Write down the truth values in the blanks.
a) (Enter A, B, C or D)
b) A B C D
Remember, with implication, the only combination that proves the statement false is T first and F second. the others are all correct.
6
Q6 3 props abstract
Consider the propositions p, q and r Fill in the gaps in the truth table below
a = b = c = d =
e = f = g = h =
i = j = k = l =
a - d - These are all opposites
e is T because I just need one of either p or q to be T
f is F because I have neither
g is T because r is F
h is T for the same reason
i is F becasue I need both not p AND not r
j is T because I have both
k is T because the first proposition is F
l is F because we have TF in that order, the only combination that returns an F
7
Q7 3 props in context
Consider the following propositions
p - Timon is singing
q - Pumba is crying
r - Nala is looking for Simba
a) Match the statements match with the following compound logic statements.
A \(\left( q\vee r \right) \wedge \neg p\) B \(r\Rightarrow \left( p\vee \neg q \right) \) C \(q\Rightarrow \left( p\vee r \right) \)
b) Fill in the gaps from the Truth Table below.
a) Enter A, B or C in these boxes
If Pumba is crying then Timon is singing or Nala is looking for Simba
Either Pumba is crying or Nala is looking for Simba, and Timon is not singing
b) Enter T or F in the bixes below
a = b = c = d =
e = f = g = h =
For part b)
a - F because we dont have either
b - T becauseat least one of q or r is true
c - T becauseat least one of p or r is true
d - F because neither are
e - F - I need both columns 4 and 5 to be true and they are not
f - T - in this case they are
g - T because q is false (TF is the only combination that returns a false result)
h - T because q is false (TF is the only combination that returns a false result)
8
Q8 Contradiction and Tautology
For a compound statement to be a tautology, all possible combinations of the propostions must be (T or F)
For a compound statement to be a contradiction, all possible combinations of the propostions must be (T or F)
For the truth tables below, say whether the statement in the last column is a Tautology (T), Contradciton (C) or Niether (N)
Enter T, C or N accordingly
\(\neg \left( p\wedge q \right) \vee q\) is
\(\left( p\vee q \right) \wedge \neg p\) is
\((p\wedge q)\Rightarrow (p\vee q)\) is
\(\neg (p\vee q)\wedge p\) is
9
Q9 Logic equivalence
Consider the following three compound logicl statements. By completing the truth table, state which two are logically equivalent.
Which Statement is logically equivalent to Statement A?
Which Statement is logically equivalent to Statement C?
10
Q10 Converse, Inverse, contrapositive
Each of the following compound logic statements is followed by its converse (A), inverse (B) and contrapositive (C) but not necesariy in that order. For each statement, enter either A, B or C next to the statements accordingly.
The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.
Question 1
First question and video
Question 2
Second question and video solution
Question 3
Third question and video solution
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