The statement (p∧q) ⇒ (p∧r) is equivalent to :
- q⇒ (p∧r)
- p⇒ (p∧r)
- (p∧r) ⇒ (p∧q)
- (p∧q) ⇒ r
The Correct Option is D
Approach Solution - 1
To determine the equivalent logical statement for \((p \land q) \Rightarrow (p \land r)\), we will use logical equivalences and analyze each option.
The implication \((p \land q) \Rightarrow (p \land r)\) can be rewritten using the definition of implication: \(A \Rightarrow B\) is equivalent to \(\neg A \lor B\).
This means:
\[ (p \land q) \Rightarrow (p \land r) \equiv \neg (p \land q) \lor (p \land r) \]
Simplifying \(\neg (p \land q) \lor (p \land r)\) involves the following steps:
- The expression \(\neg (p \land q)\) using De Morgan's Laws becomes \(\neg p \lor \neg q\).
- Thus, \(\neg (p \land q) \lor (p \land r)\) becomes \((\neg p \lor \neg q) \lor (p \land r)\).
- The distributive law allows us to further simplify this to:
- \((\neg p \lor \neg q \lor p) \land (\neg p \lor \neg q \lor r)\)
- We know that \((\neg p \lor p)\) is a tautology and simplifies to \(True\), so the expression reduces to:
- \(True \land (\neg q \lor r)\)
- Which simplifies finally to \(\neg q \lor r\)
- \(\neg q \lor r\) is equivalent to \(q \Rightarrow r\).
Therefore, the equivalent statement to \((p \land q) \Rightarrow (p \land r)\) is \((p \land q) \Rightarrow r\).
This matches the correct option: \((p \land q) \Rightarrow r\).
Approach Solution -2
| A | B | ||||||||
| P | q | r | p∧ q | p∧ r | A → B | q → B | p → B | B → A | A → r |
| T | T | T | T | T | T | T | T | T | T |
| T | F | T | F | T | T | T | T | F | T |
| F | T | T | F | F | T | F | T | T | T |
| F | F | T | F | F | T | T | T | T | T |
| T | T | F | T | F | F | F | F | T | F |
| T | F | F | F | F | T | T | F | T | T |
| F | T | F | F | F | T | F | T | T | T |
| F | F | F | F | F | T | T | T | T | T |
(p∧ q) ⇒ (p∧ r) is equivalent to (p∧ q) ⇒ r
So, the correct option is (D): (p∧q) ⇒ r
Top Questions on mathematical reasoning
- In a group of 100 people, 30 speak only English, another 50 speak only Telugu, and another 20 speak both. How many speak at least one language?
- AP LAWCET - 2025
- General Knowledge and Mental Ability
- mathematical reasoning
- Which is the odd number in the series: 2, 4, 8, 10, 16?
- AP LAWCET - 2025
- General Knowledge and Mental Ability
- mathematical reasoning
- Identify the next in the series: AZ, BY, CX, ?
- AP LAWCET - 2025
- General Knowledge and Mental Ability
- mathematical reasoning
- If CAT = 3120, then DOG = ?
- AP LAWCET - 2025
- General Knowledge and Mental Ability
- mathematical reasoning
- If today is Monday, what will be the day after 100 days?
- AP LAWCET - 2025
- General Knowledge and Mental Ability
- mathematical reasoning
Questions Asked in JEE Main exam
The equivalent resistance between the points \(A\) and \(B\) in the given circuit is \[ \frac{x}{5}\,\Omega. \] Find the value of \(x\).
- JEE Main - 2026
- Current electricity
Method used for separation of mixture of products (B and C) obtained in the following reaction is:
- JEE Main - 2026
- p -Block Elements
- Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are \(2.4 \times 10^8\) m/s and \(2.7 \times 10^8\) m/s, respectively, then the value of critical angle is :
- JEE Main - 2026
- Newtons Laws of Motion
In the following \(p\text{–}V\) diagram, the equation of state along the curved path is given by \[ (V-2)^2 = 4ap, \] where \(a\) is a constant. The total work done in the closed path is:
- JEE Main - 2026
- Thermodynamics
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
- JEE Main - 2026
- Coordinate Geometry
Concepts Used:
Mathematical Reasoning
Mathematical reasoning or the principle of mathematical reasoning is a part of mathematics where we decide the truth values of the given statements. These reasoning statements are common in most competitive exams like JEE and the questions are extremely easy and fun to solve.
Types of Reasoning in Maths:
Mathematically, reasoning can be of two major types such as:
- Inductive Reasoning - In this, method of mathematical reasoning, the validity of the statement is examined or checked by a certain set of rules, and then it is generalized. The principle of mathematical induction utilizes the concept of inductive reasoning.
- Deductive Reasoning - The principle is the opposite of the principle of induction. Contrary to inductive reasoning, in deductive reasoning, we apply the rules of a general case to a provided statement and make it true for particular statements. The principle of mathematical induction utilizes the concept of deductive reasoning.