Show if the following are logically equivalent
I) P->Q and NQ ->Np
II) N{P∆Q} AND {NP∆NQ}
2] Give A={7,8,4,5}
B={X, Y, Z}
I) What Type Of Mapping Is A TO B
II) state the D{F}
III) state the rang of R?
To determine whether two logical expressions are logically equivalent, we can use truth tables. Let's evaluate the truth values of both statements:
I) P -> Q and ¬Q -> ¬P
Here's the truth table for this statement:
| P | Q | P -> Q | ¬Q -> ¬P |
|-----|-----|--------|----------|
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
II) ¬(P ∆ Q) and (¬P ∆ ¬Q)
Here's the truth table for this statement:
| P | Q | ¬(P ∆ Q) | (¬P ∆ ¬Q) |
|-----|-----|----------|-----------|
| T | T | F | F |
| T | F | F | T |
| F | T | F | T |
| F | F | T | F |
By comparing the truth values in both truth tables for all possible combinations of P and Q, we see that both statements have the same truth values for all cases. Hence, I) and II) are logically equivalent.
Moving on to the second part of the question:
1) What type of mapping is A to B?
To determine the type of mapping from set A to set B, we need to compare the cardinalities of the two sets. Set A has 4 elements (7, 8, 4, 5), whereas set B has 3 elements (X, Y, Z). Since the cardinality of set A is greater than the cardinality of set B, the mapping from A to B cannot be one-to-one (injective). However, since the cardinality of set A is less than the cardinality of set B, the mapping is not onto (surjective) either. Therefore, the mapping from A to B is a many-to-one mapping.
2) State the domain of function F.
In the given question, there is no mention of a function F. Hence, there is no domain to state.
3) State the range of relation R.
Again, there is no mention of a relation R in the given question. Therefore, we cannot state the range of R.