Note: In this quiz, the word "or" always has inclusive meaning. That means, "A or B" is always equivalent to "either A or B or both of them".

Check all valid conclusions and do not check the invalid ones.

**1. There is at least one A. All A are B, and at least one, but not all B are C.
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All A are C.

Not all A are C.

All B are C.

Not all B are C.

All C are B.

Not all C are B.

All C are A.

Not all C are A.

All B are A.

Not all B are A.

At least one C is B.

No C is B.

At least one C is A.

No C is A.

At least one B is A.

No B is A.

At least one A is B.

No A is B.

At least one A is C.

No A is C.

All B except those that are C are A.

Not all B except those that are C are A.

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**2. Z is an element of the sets A and B. Y is an element of the sets A and C. X is an element of the sets B and C.**

Z is an element of the set A.

Z is not an element of the set A.

Z is an element of the set B.

Z is not an element of the set B.

Z is an element of the set C.

Z is not an element of the set C.

X and Y are elements of the set A.

At least one of the two, X and Y, is not an element of the set A.

X and Y are elements of the set B.

At least one of the two, X and Y, is not an element of the set B.

X and Y are elements of the set C.

At least one of the two, X and Y, is not an element of the set C.

Y is not an element of the set A.

Y is an element of the set A.

Y is not an element of the set B.

Y is an element of the set B.

Y is not an element of the set C.

Y is an element of the set C.

Y is an element of the set A or of the set B.

Y is neither an element of the set A nor of the set B.

Y is an element of the set A, but not of the set B.

Y is an element of the set B or it is not an element of the set A.

**3. P is true if and only if both Q and R are true.**

If Q is true and R is true, P is true.

If Q is true and R is true, P is false.

If Q is true and R is false, P is true.

If Q is true and R is false, P is false.

If Q is false and R is true, P is true.

If Q is false and R is true, P is false.

If Q is false and R is false, P is true.

If Q is false and R is false, P is false.

If P is true and R is true, Q is true.

If P is true and R is true, Q is false.

If P is false and R is true, Q is true.

If P is false and R is true, Q is false.

If P is false and R is false, Q is true.

If P is false and R is false, Q is false.

P may be true and at the same time Q may be false.

It is not possible that P is true and at the same time Q is false.

P may be true and at the same time R may be false.

It is not possible that P is true and at the same time R is false.

P may be false and at the same time Q may be true.

It is not possible that P is false and at the same time Q is true.

P may be false and at the same time R may be true.

It is not possible that P is false and at the same time R is true.

**4. Q is true if and only if A is true and B is false.**

If A is true and B is true, Q is true.

If A is true and B is true, Q is false.

If A is true and B is false, Q is true.

If A is true and B is false, Q is false.

If A is false and B is true, Q is true.

If A is false and B is true, Q is false.

If A is false and B is false, Q is true.

If A is false and B is false, Q is false.

If Q is true and B is false, A is true.

If Q is true and B is false, A is false.

If Q is false and B is true, A is true.

If Q is false and B is true, A is false.

If Q is false and B is false, A is true.

If Q is false and B is false, A is false.

Q may be true and at the same time A may be false.

It is not possible that Q is true and at the same time A is false.

Q may be true and at the same time B may be false.

It is not possible that Q is true and at the same time B is false.

Q may be false and at the same time A may be true.

It is not possible that Q is false and at the same time A is true.

Q may be false and at the same time B may be true.

It is not possible that Q is false and at the same time B is true.

**5. R is true if and only if B is true or A is false.**

If A is true and B is true, R is true.

If A is true and B is true, R is false.

If A is true and B is false, R is true.

If A is true and B is false, R is false.

If A is false and B is true, R is true.

If A is false and B is true, R is false.

If A is false and B is false, R is true.

If A is false and B is false, R is false.

If R is true and B is true, A is true.

If R is true and B is true, A is false.

If R is true and B is false, A is true.

If R is true and B is false, A is false.

If R is false and B is false, A is true.

If R is false and B is false, A is false.

R may be true and at the same time A may be false.

It is not possible that R is true and at the same time A is false.

R may be true and at the same time B may be false.

It is not possible that R is true and at the same time B is false.

R may be false and at the same time A may be true.

It is not possible that R is false and at the same time A is true.

R may be false and at the same time B may be true.

It is not possible that R is false and at the same time B is true.