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(p ^ q ^ r) v (p ^ q ^ ~r) v (p ^ ~q ^ r) = (p ^ q) v (p ^ r)

is this true or false?

1 Answer

1 vote

Answer: Therefore, the statement is true.

Explanation:

The given statement:

(p ^ q ^ r) v (p ^ q ^ ~r) v (p ^ ~q ^ r) = (p ^ q) v (p ^ r)

is true in classical logic. This is known as the Distributive Law of Disjunction over Conjunction, which is a valid logical equivalence.

Here's a brief explanation:

Left-hand side (LHS):

(p ^ q ^ r) v (p ^ q ^ ~r) v (p ^ ~q ^ r) represents a disjunction (OR) of three different conjunctions (AND).

Each conjunction has p as a common factor.

Right-hand side (RHS):

(p ^ q) v (p ^ r) represents a disjunction (OR) of two conjunctions (AND).

Here, p is also a common factor.

In both the LHS and RHS, the common factor p is combined with different conditions using conjunctions (AND). Since both sides have the same common factor p and use the same logical connectives, they are equivalent. Therefore, the statement is true.

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