To answer this question, we should substitute the values of x in the predicate. Once we do that, the predicate is transformed into a proposition, and we can state the truth value.

So, let’s start answering the questions.

**a) **P(0)

P(x): “x=x^{2}”

P(0): “0=0^{2}=0”

**Answer**: true.

**b) **P(1)

P(x): “x=x^{2}”

P(1): “1=1^{2}=1”

**Answer**: true.

**c) **P(2)

P(x): “x=x^{2}”

P(2): “2=2^{2}=4”

**Answer**: false.

**d) **P(−1)

P(x): “x=x^{2}”

P(-1): “-1=(-1)^{2}=1”

**Answer**: false.

**e) **∃xP(x)

To answer this type of question, we can use the answer to the previous ones.

We already know that exists at least one value x, such that P(x) is true.

Such a value can be x=0.

**Answer:** true.

**f) **∀xP(x)

In this case, we can use a similar approach to e).

From c), we know that exists at least on value x=2, such that P(x) is false.

Therefore, we can state that not for all values of x, P(x) is true.

**Answer:** false.

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