Select True Or False To Tell Whether The Following Conjunctions Are True Or False. Use The Truth Table If Needed. Is The Statement $6+3=9$ And $5 \cdot 5=20$ True Or False?A. True B. False

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Understanding Conjunctions

In mathematics and logic, a conjunction is a statement that combines two or more individual statements using the logical operator "and." The conjunction is considered true only if all the individual statements are true. If any of the individual statements are false, the conjunction is false. In this article, we will explore how to evaluate conjunctions using truth tables and apply this knowledge to a specific statement.

Truth Tables: A Tool for Evaluating Conjunctions

A truth table is a mathematical table used to evaluate the truth value of a logical statement. It is a table that lists all possible combinations of truth values for the individual statements and the resulting truth value of the conjunction. The truth table for a conjunction with two individual statements is as follows:

Statement 1 Statement 2 Conjunction
T T T
T F F
F T F
F F F

In this table, T represents true and F represents false. The conjunction is true only when both individual statements are true.

Evaluating the Statement

Now, let's apply this knowledge to the statement "$6+3=9$ and $5 \cdot 5=20$". To evaluate this statement, we need to determine the truth value of each individual statement.

Individual Statements

  1. 6+3=96+3=9

To evaluate this statement, we need to calculate the sum of 6 and 3. The result is 9, which is equal to the statement. Therefore, this statement is true.

  1. 5⋅5=205 \cdot 5=20

To evaluate this statement, we need to calculate the product of 5 and 5. The result is 25, which is not equal to the statement. Therefore, this statement is false.

Conjunction Evaluation

Now that we have evaluated the individual statements, we can use the truth table to determine the truth value of the conjunction. Since one of the individual statements is false, the conjunction is false.

Conclusion

In conclusion, the statement "$6+3=9$ and $5 \cdot 5=20$" is false. This is because one of the individual statements is false, and the conjunction is only true when both individual statements are true.

Additional Examples

To further illustrate the concept of conjunctions, let's consider a few more examples.

Example 1

Statement: "$2 \cdot 3=6$ and $4+2=7$"

Individual Statements:

  1. 2 \cdot 3=6$ (True)

  2. 5 \cdot 2=10$ (True)

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