Translate the following arguments into symbolic form. Then determine whether each is valid or invalid by constructing a truth table for each. If microchips are made from diamond wafers, then computers will generate less heat. Computers will not generate less heat and microchips will be made from diamond wafers. Therefore, synthetic diamonds will be used for jewelry.

Argument is valid.

Step-by-step explanation:

Let M, C and S represent the premises. The the statements can be represented using these letters as:

M: Microchips are made from diamond wafers.

C: Computers will generate less heat.

S: Synthetic diamonds will be used for jewelry.

1st premises:

If microchips are made from diamond wafers, then computers will generate less heat.

Notice that the above statement is a conditional statement. So to represent such relationship, the material implication is used which has ⊃ symbol.

This can be translated into symbolic form using the horseshoe operator ⊃ as:

M ⊃ C

2nd premises:

Computers will not generate less heat and microchips will be made from diamond wafers.

Notice that the above statement is a compound statement. It has two part joined together with "and" i.e.  Computers will not generate less heat "and" microchips will be made from diamond wafers. These are two conjuncts and   both should be true for the premises/compound statement to be true. So to represent such relationship, the Conjunction is used which has • symbol.

This can be translated into symbolic form using the dot • symbol as:

Computer will not generate less heat is represented as ~C because C states that Computer will generate less heat and ~C states that Computer will not generate less heat. So ~C is the negation of C. Now the complete symbolic form is:

~C • M

third premises:

Therefore, synthetic diamonds will be used for jewelry is represented as:

S

This is basically the conclusion statement.

Truth Tables:

For M:

M

T

T

T

T

F

F

F

F

For C:

C

T

T

F

F

T

T

F

F

For ~C:

It reverses the truth values of C

~C

F

F

T

T

F

F

T

T

For S:

S

T

F

T

F

T

F

T

F

For M ⊃ C:

The above mentioned premise formed with this connective is true unless the left the antecedent is true and right the consequent is false.

M       C        M ⊃ C

T         T             T

T         T             T

T         F             F

T         F             F

F         T             T

F         T             T

F         F             T

F         F             T

For ~C • M

The truth table of conjunction of ~C and M is formed using truth table of reverse of C i.e.  ~C and that of M. The conjunction is true when both conjuncts are true and if either of the two conjuncts is false then whole conjunction is false.

M       ~C        ~C • M

T         F             F

T         F             F

T         T             T

T         T             T

F         F             F

F         F             F

F         T             F

F         T             F

Now combining all to check the validity:

M       C       S        M ⊃ C     ~C • M             S

T         T        T             T               F                T

T         T         F            T               F                F

T         F         T            F               T                T

T         F         F            F               T                F

F         T         T            T               F                T

F         T         F            T               F                F

F         F         T            T               F                T

F         F          F           T               F                 F

The given argument is valid. The is because it is impossible for the above premises to be true and the conclusion to be false. You can see that when the premises M ⊃ C  and ~C • M are true the conclusion S is also true.

We check validity by checking in the truth table if there is a row that has all true premises and a false conclusion. If there is then argument is invalid. Here in the truth table you can see that no row has all the premises true and a false conclusion. So the argument is valid.

your answer is a

step-by-step explanation:

4*4*4*4=256