Lesson 2
Conjunction
At the end of this lesson, the learner should be able
to ● identify and give examples of a conjunction; ●
construct truth tables for conjunctions; and
● determine the truth value of a conjunction with two
or more propositions.
● How do we define a conjunction? ●
What makes a conjunction true or false? ●
How do we construct truth tables?
For this lesson, we will discuss conjunctions. Before we
proceed, let us read this story named Linda the Bank
Teller.
(Click on the link below to show the story.)
Brogaard, Berit. 2016. “Linda The Bank Teller Case Revisited”.
Retrieved 17 May 2019 from https://bit.ly/2wwOLWz
Is there a relation between
conjunctions and fallacies?
● Did you also make the same mistake in committing a
conjunction fallacy?
● Do you have a hard time deciphering whether individual
statements are true or false because they are coupled with
other statements?
● What is a conjunction? Which among the two statements
about Linda is a conjunction?
Truth tables
1
represent the relationship between the truth values of propositions and
compound propositions formed from those propositions.
Example:
We may start by creating a truth table for the negation of
any proposition ��.
�� ~��
T F
F T
Truth tables
1
represent the relationship between the truth values of propositions and
compound propositions formed from those propositions.
F T
Example:
The first column contains the two possible truth values of
proposition ��. The second column reflects the truth value
of ~�� corresponding to the truth value of ��.
Conjunction
2
a proposition formed by combining two propositions (called conjuncts) with the
word and. The connective “and” implies the idea of “both.” For the conjunction
�� ∧ �� to be true, both �� and �� must be true.
Example:
Below is the truth table for �� ∧ ��.
�� �� �� ∧ ��
T T T
T F F
F T F
F F F
Conjunction
2
a proposition formed by combining two propositions (called conjuncts) with the
word and. The connective “and” implies the idea of “both.” For the conjunction
�� ∧ �� to be true, both p and q must be true.
Example:
Observe that the first two columns contain all the possible
combination of truth values of �� and ��.
We started with having both T as their individual truth values
and ended with having both F.
Conjunction
2
a proposition formed by combining two propositions (called conjuncts) with the
word and. The connective “and” implies the idea of “both.” For the conjunction
�� ∧ �� to be true, both p and q must be true.
Example:
The third column reflects that �� ∧ �� is only true when
both �� and �� are true.
Example 1: Let �� represent “1 < 3” and �� represent “1
> 3.” Find the truth value of �� ∧ ��.
Example 1: Let �� represent “1 < 3” and �� represent “1
> 3.” Find the truth value of �� ∧ ��.
Solution:
The statement 1 < 3 is true. Hence, �� is
true. The statement 1 > 3 is false. Hence, ��
is false.
Since one of the two statements if false, �� ∧ �� is false.
Example 2: Construct the truth table for the compound
proposition �� ∧ ~��.
Example 2: Construct the truth table for the compound
proposition �� ∧ ~��.
Solution: First, write all the possible combinations of truth
values of �� and ��.
�� ��
T T
T F
F T
F F
Example 2: Construct the truth table for the compound
proposition �� ∧ ~��.
Solution: Second, get the negation of ��.
�� �� ~��
T T F
T F T
F T F
F F T
Example 2: Construct the truth table for the compound
proposition �� ∧ ~��.
Solution: Finally, determine the truth values of �� ∧ ~��.
�� �� ~�� �� ∧
~��
T T F F
T F T T
F T F F
F F T F
Individual Practice:
1. Construct the truth table for the conjunction ~�� ∧ �� ∧
��.
2. Consider the following propositions ��, ��,
and ��. ��: A square’s interior angles sum
up to 360°.
��: A rectangle is a parallelogram.
Find the truth value of each of the following
conjunctions: a. �� ∧ ��
b. ~�� ∧ ��
c. ~�� ∧ ~��
Group Practice: To be done in 2 to 5 groups
Consider the following situation:
In a college, students are randomly selected. A certain
number like tea, a certain number like coffee, and a certain
number like both tea and coffee.
Let �� be the students who like tea.
Let �� be the students who like coffee.
Show in a Venn diagram where is �� ∧ ��, ~�� ∧ ��, ��
∧ ~��, and ~�� ∧ ~��. Explain and translate these into
statements describing the situation.
Truth tables
1
represent the relationship between the truth values of propositions and
compound propositions formed from those propositions.
Conjunction
2
a proposition formed by combining two propositions (called conjuncts) with the
word and. The connective “and” implies the idea of “both.” For the conjunction
�� ∧ �� to be true, both p and q must be true.
● How do we construct truth tables?
● How can we assess the truthfulness of multiple statements
inside a story?
● What do you call propositions joined by the word “or”?
