1. Idea of Compound Statements
A compound statement is formed by combining two or more simple statements using logical connectors. These connectors express relationships such as "and", "or", "if…then", and "if and only if". Compound statements allow us to describe more complex situations using logic.
Each simple statement in the combination has its own truth value, and the truth of the compound statement depends on how these values interact through the connectors.
1.1. Why compound statements are needed
Most real statements are not isolated facts. They involve conditions, combinations, or choices. Compound statements capture these relationships clearly and help in reasoning and proving mathematical results.
2. Conjunction (AND)
A conjunction joins two statements using the word and. It is written as "P and Q". The conjunction is true only when both statements are true.
2.1. Examples
- P: "n is even"
- Q: "n is greater than 10"
- Compound: "n is even and n is greater than 10"
This is true only when both parts hold.
3. Disjunction (OR)
A disjunction joins two statements using the word or. In mathematics, "or" is usually taken as inclusive, meaning P or Q or both.
The disjunction "P or Q" is false only when both P and Q are false.
3.1. Examples
- P: "x is greater than 5"
- Q: "x is even"
- Compound: "x is greater than 5 or x is even"
4. Implication (If–Then)
An implication has the form "If P, then Q". It expresses that whenever P holds, Q must also hold. This type of statement is fundamental in mathematical arguments and proofs.
The implication is considered false only when P is true and Q is false. In all other cases it is taken as true.
4.1. Examples
- P: "a number is even"
- Q: "the number is divisible by 2"
- Statement: "If a number is even, then it is divisible by 2"
5. Bi-conditional (If and Only If)
A bi-conditional combines two implications. It has the form "P if and only if Q" and is written as "P ⇔ Q". It means both "If P then Q" and "If Q then P" hold.
A bi-conditional is true when P and Q have the same truth value: either both true or both false.
5.1. Examples
- P: "a number is even"
- Q: "the number is divisible by 2"
- Bi-conditional: "A number is even if and only if it is divisible by 2"
6. Truth Values of Compound Statements
Although truth tables give a complete picture, compound statements can often be understood intuitively by thinking about how their components interact. The connector determines how the truth values combine.
6.1. Quick intuition
- "P and Q" → both must be true
- "P or Q" → at least one is true
- "If P then Q" → false only when P is true and Q is false
- "P if and only if Q" → true when P and Q match
7. Worked Examples
These examples show how compound statements are formed and interpreted.
7.1. Example 1: AND
Statements:
- P: "7 is a prime number"
- Q: "7 is even"
Compound: "7 is prime and 7 is even" → false, because Q is false.
7.2. Example 2: OR
Statements:
- P: "4 is even"
- Q: "4 is greater than 10"
Compound: "4 is even or 4 is greater than 10" → true (since P is true).
7.3. Example 3: Implication
Statement: "If 3x = 12, then x = 4" → true.
Here, whenever the condition (3x = 12) holds, the conclusion must hold.
7.4. Example 4: Bi-conditional
Statement: "x = 2 if and only if x² = 4" → false.
The backward direction fails because x² = 4 also holds for x = -2.
8. Notes and Observations
Some useful points:
- Compound statements combine simpler statements using logical connectors.
- Understanding how each connector behaves is essential for correct reasoning.
- Implications and bi-conditionals often appear in proofs.
- Disjunction in mathematics is usually inclusive (P or Q or both).