## Conjunction in Discrete MathematicsLogical reasoning is an important skill that can be used in various fields like engineering, science, and as well as our daily life activities. We can also use logical reasoning in mathematical problem-solving strategies. With the help of logical reasoning and given facts, we can quickly get a conclusion. There are various kinds of logical connections to solve problems in mathematics. Some of them, the commonly used connectives are described as follows: - Implication
- Negation
- Disjunction
- Equivalent
- Conjunction
In this section, we are going to learn about one of the connectives called Conjunction. We will learn the definition, examples, rules, and truth tables of conjunction. ## Definition of ConjunctionWhen we add the two statements with the help of 'AND' symbol, then it will be known as conjunction. For conjunctions, the combined compound statement will be true if and only if both statements are true. Suppose there are two statements, i.e., "circles are curves" and "parallelograms are rectangle". Now we will make a compound statement by combining both statements with 'and' like this "circles are curves and parallelograms are rectangle". When both the statements are true, only then this newly created compound statement will be true. The compound statement will be false if only one statement is true and another one is false.
## Conjunction in Discrete mathematicsThe conjunction can be described as a statement, which can be formed by adding two statements with the help of connector AND. The symbol ∧ is used for the conjunction. We can read this symbol as "and". If two statements, x, and y are joined in a statement, then the conjunction can be indicated symbolically in the form of x ∧ y. This statement will be true when both the combined statements are true. In all the other cases, it will be false. The conjunction of x and y is shown in the following image: ## Rules of Conjunction- If both the combined statements are true, only then the conjunction statement will be true. In all the other cases, it will be false.
- This process is very similar to AND gate, which is covered by the topic Gate logic.
- The symbol ∧ is used to indicate the conjunction. We can read this symbol as "and", which is a type of logical connective.
- We normally use the alphabetical letters to represent the statements. Suppose there are two statements, X and Y. When we add a conjunction connector, then the statement will become a compound statement. In this case, the compound statement is represented as X ∧ Y, and it will be read as "X and Y".
## Conjunction Truth tableWith the help of truth table, we can understand the final value of the compound statement, which is based on the values of individual statements. In the following truth table, all the possible combinations are represented. In the truth table, a true value is indicated with the help of letter 'T', and a false value is indicated with the help of letter 'F'. In the following truth table, we have two statements, X and Y, and a compound statement X ∧ Y. Now, we have to make a truth table of these statements.
With the help of above table, we can see that when both the statements X and Y are true, only then the conjunction statement (X ∧ Y) is true. The conjunction statement is false when one of them is not true or false. In other words, we can say that the conjunction will be false in all the cases except when both the statements are true. ## Examples of ConjunctionThere are a lot of examples of conjunctions, which are described as follows:
So we can see that when X and Y are true, only then the conjunction X ∧ Y is true. So the conjunction of both the statements is true.
If n = 3, in this case, p is true, and q is true. So the conjunction p ∧ q is also true. If n = 9, in this case, p is true, and q is false. So the conjunction p ∧ q is false. If n = 2, in this case, p is false, and q is true. So the conjunction p ∧ q is false. If n = 6, in this case, p is false, and q is false. So the conjunction p ∧ q is also false.
- p and q
- ~p and q
- ~q and p
The truth table for conjunction ~p and q is described as follows:
The truth table for conjunction ~q and p is described as follows:
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