Logic and Conditional Statements Quiz

In formal logic, a conditional statement 'If P then Q' is logically equivalent to its contrapositive 'If not Q then not P.' Here, P is 'Yoda cannot use a lightsaber' and Q is 'Yoda cannot help Luke.' The contrapositive is 'If Yoda CAN help Luke (not Q), then he CAN use a lightsaber (not P)' - which is option B. Options A and D assert conjunctions, not conditionals. Option C claims a biconditional (if and only if) which is stronger than the original.

To solve this question, the user needs to understand logical connectives and how to interpret compound statements.

The statement "Shelly likes Mike and she likes John" is a compound statement that consists of two individual statements connected by the logical connective "and." The truth value of this compound statement depends on the truth values of its individual statements.

Since the question states that Shelly does not like John, the individual statement "Shelly likes John" is false. Therefore, the overall truth value of the compound statement is false, regardless of whether the statement "Shelly likes Mike" is true or false.

Now, let's go through each option:

A. True: This option is incorrect because the compound statement is false.

B. False: This option is correct. As explained above, the compound statement is false due to the falsehood of the individual statement "Shelly likes John."

C. Shelly likes Peter: This option is not relevant to the given statement.

D. none of the above: This option is incorrect because option B is the correct answer.

Therefore, The Answer is: B. False.

If y = 3: (a) 3 is prime and odd (True). (b) 3 is odd (True), so the 'or' statement is True. (d) 3 is odd and 6 is even (True). (c) 'y is not prime' is False, and 'y is odd' is True; 'False and True' results in False.

The contrapositive of 'If P then Q' is 'If not Q then not P,' and these are logically equivalent. Here P is 'today is Sunday' and Q is 'Matt cannot play hockey.' The contrapositive becomes 'If Matt CAN play hockey (not Q), then today is NOT Sunday (not P)' - option B. Option C merely restates the original. Options A and D are not equivalent.

The statement 'x + x = 3x' simplifies to '2x = 3x', which is only true if x = 0. An 'if and only if' statement requires both sides to have the same truth value. '2x - x = 2x' simplifies to 'x = 2x', which is also only true if x = 0. Thus, they are logically equivalent.

An open sentence is a statement containing a variable (like 'It' or 'x') that can be true or false depending on what the variable represents. 'It was blue...' is open because 'It' is an undefined pronoun. The other options are closed statements with specific truth values.

A conditional 'If P, then Q' is false only when P is true and Q is false. If x = 32, it is divisible by 8 (true), but it is not divisible by 6 (false). For x = 48, both are true; for x = 6, P is false.

Using the Principle of Inclusion-Exclusion: Total students in music or art = 18 (music) + 26 (art) - 2 (both) = 42. To find those in neither, subtract from the total class size: 50 - 42 = 8.

The original statement is true - all right angles measure 90 degrees, so any two right angles have equal measure. However, its converse is false - two angles can have equal measure without being right angles (they could both be 30 degrees, 45 degrees, etc.). This demonstrates that a true conditional does not guarantee a true converse.

The statement 'divisible by 2 OR divisible by 3' is false only when BOTH conditions are false. We need a number not divisible by 2 (not even) AND not divisible by 3. Checking options: 6 is divisible by both (true), 8 is divisible by 2 (true), 15 is divisible by 3 (true), but 11 is not divisible by 2 (it's odd) and not divisible by 3 (1+1=2, not divisible by 3). Only 11 makes the OR statement false.

The contrapositive of 'If P then Q' is 'If not Q then not P,' and these are logically equivalent. Here P is 'Tom buys a red skateboard' and Q is 'Amanda buys green in-line skates.' The contrapositive becomes 'If Amanda does NOT buy green skates (not Q), then Tom does NOT buy a red skateboard (not P)' - option A. Option B is the inverse (not equivalent). Option C is the converse (not equivalent). Option D contradicts the original.

The inverse of a conditional statement 'If P, then Q' is 'If not P, then not Q'. For the statement 'If it is snowing (P), then Skeeter wears a sweater (Q)', the inverse is 'If it is not snowing, then Skeeter does not wear a sweater.'

The converse of a conditional statement 'If P, then Q' is 'If Q, then P'. Given 'If x > 4 (P), then x + 2 > 5 (Q)', the converse is 'If x + 2 > 5, then x > 4'. Other options represent the inverse or contrapositive.

The logical 'or' statement is true if at least one condition is met. For x = 1, the condition x < 3 is satisfied (1 < 3), making the whole statement true. For x = 3, 4, or 5, neither x > 5 nor x < 3 is true.