For three simple statements p, q, and r, p → (q ˅ r) is logically equivalent to
For three simple statements p, q, and r, p → (q ˅ r) is logically equivalent to
- (p ˅ q) → r
- (p → ∼q) ˄ (p → r)
- (p → q) ˅ (p → r)
- (p → q) ˄ (p → ∼r)
The Correct Option is C
Solution and Explanation
To determine the logical equivalence of the statement p → (q ˅ r), we can simplify it using logical equivalences and compare it to the given options.
Starting with p → (q ˅ r), we can apply the distributive law:
p → (q ˅ r) ≡ (p → q) ˅ (p → r).
Comparing this result to the given options, we can see that the statement (C) (p → q) ˅ (p → r) is equivalent to p → (q ˅ r).
Therefore, the correct option is (C).
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Concepts Used:
Statements
A statement is a sentence that is either true or false, but not both true and false simultaneously.
Types of Statements:
Simple Statement
- If a statement cannot be further broken down into various statements, or in simpler words if it is concrete by itself, it is called a Simple Statement.
- Examples include:
- A kite is not a rhombus.
- 15 is an odd number.
Compound Statement
- If a statement can further be broken down into simpler statements so that from a main statement, we can yield more than one statement, then it is called a Compound Statement.
- Consider the statement “10 is non-negative and a multiple of 5” which can be broken down into the statements: “10 is non-negative” and “10 is a multiple of 5”.
If-Then Statements
- If we encounter an if-then statement i.e. ‘if a then b’, then by proving that a is true, b can be proved to be true or if we prove that b is false, then a is also false.
- If we encounter a statement which says ‘a if and only if b’, then we can give reason for such a statement by showing that if a is true, then b is also true and if b is true, then a is also true.
- Example:
- a: 8 is multiple of 64
- b: 8 is a factor of 64
Since one of the given statements i.e. a is true, therefore, a or b is true.