A conjunction is a statement formed by adding two statements with the connector AND. The symbol for conjunction is ‘∧’ which can be read as ‘and’. When two statements p and q are joined in a statement, the conjunction will be expressed symbolically as p∧q. If both the combining statements are true then this statement will be true otherwise, it is false.
Rules for a conjunction to be true
- The conjunction statement will only be true if both the combining statements are true otherwise, the disjunction is false.
- It is similar to an AND gate which is utilized under the topic Gate logic.
- The definition is stated as let p and q be the two statements. The compound statement p ∧ q is called as the conjunction of p and q.
- The symbol “∧” that denotes the conjunction is called as “and” which is the logical connective.
Conjunction Truth Table
| P | Q | P ∧ Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
In this table, we can say that the conjunction is true only when both P and Q are true. If they are not then the conjunction statement will be false.
Solved Problems:
Example 1: Let r: 5 be a rational number and s: 15 be a prime number. Is it a conjunction?
Solution:
Given that r: 5 is a rational number. This proposition is true.
s: 15 is a prime number. This proposition is false as 15 is a composite number.
Therefore, as per the truth table, r and s is a false statement.
So, r ∧ s = F
Example 2: Let a: x is greater than 9 and b: x be a prime number. Is it a conjunction?
Solution:
Since x is a variable whose value we don’t know. Let us define a range for a and b.
To find the range let us take certain values for x;
When x= 6: a is false so is b hence a ∧ b is false.
When x= 3: a is false but b is true. But still, a ∧ b is false.
When x= 10: a is true but b is false. But still, a ∧ b is false.
When x= 11: a is true and b is true. Hence, a ∧ b is true.
Hence the conjunction a and b is only true when x is a prime number greater than 9.
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