Consider the following four statements. Statement 1: "Statement 3 is true." Statement 2: "Statement 1 is true" Statement 3: "Statement 1 is true and Statement 2 is false" Statement 4: "Statements 1, 2 and 3 are false" Which of the above statements must be true for the four statements to be mutually consistent?
Statement Consistency Analysis
This problem involves analyzing logical consistency among four statements, where the truth value of each statement depends on the truth values of others.
Understanding the Statements
Let T represent True and F represent False. We can translate the statements into logical dependencies:
- Statement 1 (S1): "Statement 3 is true." (S1 implies S3 is T)
- Statement 2 (S2): "Statement 1 is true." (S2 implies S1 is T)
- Statement 3 (S3): "Statement 1 is true and Statement 2 is false." (S3 implies S1 is T AND S2 is F)
- Statement 4 (S4): "Statements 1, 2 and 3 are false." (S4 implies S1 is F AND S2 is F AND S3 is F)
Deductive Reasoning Process
We test the truth value of each statement by assuming it is true and checking for contradictions. If an assumption leads to a contradiction, the statement must be false.
-
Assume S1 is True:
- If S1 is T, then S3 must be T (from S1's content).
- If S3 is T, then S1 must be T and S2 must be F (from S3's content).
- So far: S1=T, S3=T, S2=F.
- Now check S2: S2 states "S1 is true". Since S1=T, S2 should be T.
- This contradicts S2=F derived from S3.
- Therefore, the assumption S1=T leads to a contradiction. S1 must be F.
-
Assume S2 is True:
- If S2 is T, then S1 must be T (from S2's content).
- However, we just proved S1 must be F.
- This is a contradiction.
- Therefore, the assumption S2=T leads to a contradiction. S2 must be F.
-
Assume S3 is True:
- If S3 is T, then S1 must be T and S2 must be F (from S3's content).
- However, we proved S1 must be F.
- This is a contradiction.
- Therefore, the assumption S3=T leads to a contradiction. S3 must be F.
-
Evaluate S4:
- We have deduced: S1=F, S2=F, S3=F.
- Statement 4 (S4) claims: "Statements 1, 2 and 3 are false."
- Since S1, S2, and S3 are indeed all False, the content of S4 is accurate.
- This means S4 must be True.
Conclusion on Truth Values
The only mutually consistent assignment of truth values is:
- Statement 1: False
- Statement 2: False
- Statement 3: False
- Statement 4: True
For the four statements to be mutually consistent, Statement 4 must be true.
Important Questions from Statement and Conclusion (Notes)
- In the following question, a statement followed by 2 conclusions is given.
Choose right conclusion which logically follows the given statement.
Statement : Morning walks are good for health.
Conclusions : I. All healthy people go for morning walk.
II. Evening walks are harmful. - If there is rainfall in a village, flowers will bloom. If flowers bloom, celebrations will definitely take place. Flowers did not bloom. Which of the following is certain?
- Study the following question carefully and choose the right answer
Statement: Should articles of only deserving authors be allowed to be published?
Arguments:
(I) No, it is not possible to draw a line between the deserving and the undeserving.
(II) Yes, it will save a lot of paper which is already in short supply.
In light of the above statements, choose the most appropriate answer from the options given below. - Statements: Morning walks are good for health.
Conclusions: I: All healthy people go for morning walks.
II: Evening walks are harmful.
Choose the correct option : - Given below are two statements related to evaluation system.
Statement (I)Formative evaluation is administered during the lesson taught or instruction imported
Statemet (II)Summative evaluation summarizes the final progress of the students as a result of a course of learning unit.
Select the correct option
