Applications of Algebra

Talk for Learning, Group work. Learning Experience: Learners in pairs discuss and construct compound and simple statements.
Activity 1: Compound and simple statements Learners in groups discuss compound statements with examples.
Example 1 A compound statement is a group of two or more statements connected using words such as 'or', 'and', 'if then', and 'if and only if'.
Each statement of a compound statement is a component statement, which can be clearly decided as a true or false statement.
The individual statements are represented as p, q, and the compound statements are represented as p v q, p ^ q, p ⇒ q, p ⇔ q.
Examples of Compound Statements:
- The grass is green and the sky is blue.
- It is cold or it is sunny.
- If a person is kind then he is helpful.
- The number 12 is an even number if and only if it is divisible by 2.
Compound statements are generally formed from simple statements, which are represented as p, q, and the compound statements are represented as p v q, p ^ q, p ⇒ q, p ⇔ q. The symbols used to connect the statements p, q are v, ^, ⇒, ⇔ represent the words 'or', 'and', 'if then', 'if and only if', and are referred to as connectives.
The words 'or', 'and' are useful to form a compound statement, but every statement having these words 'or', 'and' need not be a compound statement.
Example 2: For the compound statement: If it is raining, then it will be very cold", write the converse statement.
Solution: Conditional Statement: P → Q: If it is raining, then it will be very cold.
Converse Statement: Q → P: If it is very cold, then it will be raining.
Example 3: What is the compound statement that can be formed from the statements P: You go regularly to school, and Q: You get good marks?
Solution: The two given statements are:
P: You go regularly to school.
Q: You get good marks.
The four possible connectives that can be used here are and, or, if... then, if and only if. Let us form the four compound statements.
Conjunction Statement: (And connective) You go regularly to school and you get good marks.
Disjunction Statement: (Or Connective) You go regularly to school or you get good marks.
Conditional Statement: (If then connective) If you go regularly to school then you get good marks.
Biconditional Statement: (If and only if connective) You go regularly to school if and only if you get good marks. Among the four statements, the conditional statement works well, as the second statement is dependent on the first statement.
Example 4: From the following pair of statements, form a compound statement using "and" and determine whether the compound statement is true or false.
- P: A square has four sides; Q: A triangle has three sides
- P: 6 + 7 = 13; Q: 5 + 4 > 8
- P: 4 ∈{primes}, Q: 4 ∈{even numbers}.
Solution P˄Q: A square has four sides, and a triangle has three sides. P˄Q is true because P and Q are both true.
P˄Q: 6 + 7 = 13 and 5 + 4 > 8. P˄Q is true because both P and Q are true.
P˄Q: 4 ∈{primes}∩ {even numbers} P ˄Q is false since P is false.
Teaching and Learning Resources:
- SHS Curriculum, Graph boards, mathematical set, ICT tools
Assessment (3.1.2.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.