Applications of Algebra

Talk for Learning, and Group work.
Learning Experience: Learners in squares discuss and construct truth tables of compound statements.
Activity 1: Learners in groups discuss various types of compound statements and their connectives.
Example 1 Disjunction Statement: The connective used for two simple statements to form a compound statement, which is disjunction, is 'OR.' In a disjunction statement, any one of the statements must be true for the disjunction statement to be true. The two simple statements represented as P and Q can be connected using OR connective and are written as P V Q. Here, any of the two statements should be true for the compound statement to be true.
Conjunction Statement: The compound statement of conjunction uses the connective 'AND' for connecting two simple statements. For this compound statement, both statements must be true for the compound statement to be true. The two simple statements P and Q can be connected using the 'And' connective, and the compound statement can be written as P ^ Q. For a conjunction compound statement, both statements should be true for the compound statement to be true.
Conditional Statement: The connective used for a conditional statement is if then. If Reema does well in the test, then she will be promoted to the next class. Here, the first statement, P, can be taken as the hypothesis, and the second statement, Q, can be taken as the conclusion. We can write condition statements of these two simple statements P, Q as If P then Q. The conditional compound statement does not hold true if the hypothesis is true and the conclusion is false. But in all other situations, the conditional statement is true.
Bi-Conditional Statement: The bi-conditional statement uses the connective 'If and only if,' which is represented by the symbol ⇔. The two statements, P and Q, are represented as a compound statement, P ⇔ Q, and here, the first statement, P, is called the antecedent, and the second statement, Q, is called the consequent. Here, the bi-conditional compound statement is true if both statements are either true or both are false.
Activity 2: Learners in groups construct the truth table of compound statements.
Truth Tables of a Compound Statement: The truth value of a compound statement depends on the truth value of the individual statements and also on the connective used to form the compound statement.
Example 1 Disjunction Truth Table uses the connective 'or' to form the compound statement. Here, even if one of the individual statements is true, then the compound statement also holds true.
P Q P V Q T F T T T T F T T F F F
Conjunction Truth Table uses the connective 'and' to form the compound statement. Here, the compound statement is true only if both the individual statements are true. Even if one of the individual statements is false, then the compound statement is considered a false statement.
P Q P ^ Q
T F F T T T F T F F F F
Conditional Truth Table uses If-then connective, which is represented as ⇒. Here, the statement P is referred to as a hypothesis, the statement Q is referred to as a conclusion. The compound statement is true if the conclusion is true, irrespective of the hypothesis. Also, the compound statement is true if both the hypothesis and the conclusion are false.
P Q P ⇒ Q
T F F T T T F T T F F T
Bi-conditional Truth Table uses the connective 'if and only if' and is represented as ⇔. Here, the first statement, P, is referred to as antecedent, and the second statement, Q, is referred to as
consequent. The bi-conditional compound statement is true if the second statement, the consequent, is false. P Q P ⇔ Q T F T T T F F T F F F T
Activity 3: Valid and invalid arguments using the truth table Learners in groups discuss valid and invalid arguments using the truth table.
Notes: A row of the truth table in which all the premises are true is called a critical row. If there is a critical row in which the conclusion is false, then it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid. If the conclusion in every critical row is true, then the argument form is valid.
Example 1 p → q ∨ ∼r q → p ∧ r
- p → r
Solution: 

The truth table shows that even though there are several situations in which the premises and the conclusion are all true (rows 1, 7, and 8), there is one situation (row 4) where the premises are true and the conclusion is false.
Teaching and Learning Resources:
- SHS Curriculum, Graph boards, mathematical set, ICT tools
Assessment (3.1.2.AS.4). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.