Four cards lie on a table. Each card has a number printed on one side and a colour on the other. The faces visible on the cards are 2, 3, red, and blue. Proposition: If a card has an even value on one side, then its opposite face is red.
2, blue
Cards Verification: Understanding the Proposition
This question asks us to identify which cards must be turned over to verify a specific proposition. The proposition states: "If a card has an even value on one side, then its opposite face is red." This is a classic logic puzzle that tests understanding of conditional statements.
Proposition Analysis: If P, Then Q
Let's break down the given proposition into a standard conditional statement format:
- Let \(P\) be the condition: "A card has an even value on one side."
- Let \(Q\) be the result: "Its opposite face is red."
So, the proposition is: "If \(P\), then \(Q\)" (written as \(P \Rightarrow Q\)).
To verify such a statement, we need to look for situations that could potentially prove it false. A conditional statement \(P \Rightarrow Q\) is only false when \(P\) is true AND \(Q\) is false. In all other cases (P false, Q true; P false, Q false; P true, Q true), the statement holds true.
Examining Each Card for Verification
We are given four cards with visible faces: 2, 3, red, and blue. Let's analyze each one:
Card 1: Visible Face '2'
- What we see: The number 2.
- Analysis: The visible face '2' means that the condition \(P\) ("A card has an even value on one side") is true for this card.
- Verification needed: According to the proposition \(P \Rightarrow Q\), if \(P\) is true, then \(Q\) (the opposite face is red) must also be true. To verify this, we absolutely must turn over this card to check the color of its opposite face. If the opposite face is not red (e.g., it's blue), then the proposition is immediately proven false.
- Conclusion: We must turn over the card with '2'.
Card 2: Visible Face '3'
- What we see: The number 3.
- Analysis: The visible face '3' means that the condition \(P\) ("A card has an even value on one side") is false for this card, as 3 is an odd number.
- Verification needed: If \(P\) is false, the conditional statement \(P \Rightarrow Q\) is always considered true, regardless of what \(Q\) is. For example, "If it rains (false), then the sun shines (true)" is not a contradiction. Turning this card over to see its color (its \(Q\)) will not help us verify or falsify the proposition. Whether the other side is red or blue, the proposition still holds because the "even value" condition was not met.
- Conclusion: We do not need to turn over the card with '3'.
Card 3: Visible Face 'red'
- What we see: The color red.
- Analysis: The visible face 'red' means that the result \(Q\) ("Its opposite face is red") is true for this card.
- Verification needed: If \(Q\) is true, the conditional statement \(P \Rightarrow Q\) is always considered true, regardless of what \(P\) is. For example, "If it rains (true), then the ground is wet (true)" holds. "If it rains (false), then the ground is wet (true)" also holds. If we turn this card over, we might find an even number (P is true) or an odd number (P is false). In both cases, the proposition remains valid. We are only concerned if \(P\) is true and \(Q\) is false. This card's visible face already confirms \(Q\) is true.
- Conclusion: We do not need to turn over the card with 'red'.
Card 4: Visible Face 'blue'
- What we see: The color blue.
- Analysis: The visible face 'blue' means that the result \(Q\) ("Its opposite face is red") is false for this card.
- Verification needed: This is a critical case. The only way the proposition \(P \Rightarrow Q\) can be false is if \(P\) is true and \(Q\) is false. We already know \(Q\) is false (because the card is blue, not red). Therefore, to ensure the proposition is true, we must check that \(P\) (the number on the other side) is also false. In other words, the number on the other side must not be an even value. If we turn this card over and find an even number, it would immediately falsify the proposition.
- Conclusion: We must turn over the card with 'blue'.
Summary of Required Card Turns
To conclusively verify the proposition "If a card has an even value on one side, then its opposite face is red," we must turn over any card that could potentially violate this rule.
| Visible Card Face | Condition/Result Status | Reason to Turn Over? |
|---|---|---|
| 2 | \(P\) is true (even number) | YES, must check if \(Q\) (red) is true. If not, proposition is false. |
| 3 | \(P\) is false (odd number) | NO, if \(P\) is false, \(P \Rightarrow Q\) is always true, regardless of \(Q\). |
| red | \(Q\) is true (red color) | NO, if \(Q\) is true, \(P \Rightarrow Q\) is always true, regardless of \(P\). |
| blue | \(Q\) is false (not red) | YES, must check if \(P\) (even number) is false. If \(P\) is true, proposition is false. |
Based on this analysis, the cards that MUST be turned over are the one showing '2' and the one showing 'blue'.
Important Questions from Verbal Ability
The intake pipes of nuclear plants and water plants along Great lakes are clogged by a certain variety of mussels that proved to be causing nuisance to the working systems. But it is found that the bags of these calms when suspended in the effluent streams of chemical plants, not only improves quality of water significantly but also remove some hazardous wastes from effluents just by feeding voraciously on the algae that they filter from water that passes by it.
Which one of the inferences given below, if true, is most strongly supported on the basis of information given in the above paragraph?
Complete the sentence choosing the right option:
The governor ______ that it would be a state holiday.
Directions: Which is the appropriate phrasal verb for the word "postpone" in the following sentence?
The Chairman had to postpone the meeting.
"The increased consumption of leafy vegetables in the recent months is a clear indication that the people in the state have begun to lead a healthy lifestyle"
Which of the following can be logically inferred from the information presented in the above statement?
The James Webb telescope, recently launched in space, is giving humankind unprecedented access to the depths of time by imaging very old stars formed almost 13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in space may even shed light on the existence of dark matter. Dark matter is supposed to interact only via the gravitational interaction and not through the electromagnetic-, the weak- or the strong-interaction. This may justify the epithet “dark” in dark matter.
Based on the above paragraph, which one of the following statements is FALSE?
