Question 4 Which of the following are propositions Give the

Question 4.

Which of the following are propositions? Give the truth values of the propositions.

Solution

(a) Consider the given statement

x² = x x

The truth value of the statement is false( F ) since 2 but 2² 2

(b) Consider the given statement

x² = x for some x

The truth value of the statement is True( T ) since 1 but 1² = 1 and 0 but 0² = 0.

(c) Consider the given statement
      x² = x . The truth value of the statement is false( F ).

(d) Consider the given statement

x² = x for exactly one x

The truth value of the statement is false( F ). since 1 but 1² = 1 and 0 but 0² = 0.

(e) Consider the given statement

xy = xz implies y = z.

The truth value of the statement is false( F ) .

If x , y and z are matrices.

(f) Consider the given statement

xy = xz implies y = z x , y , z

The truth value of the statement is True( T ) .

(g) Consider the given statement

w₁w₂ = w₁w₃ implies w₂ = w₃ for all words w₁ ,w₂ ,w₃ ^*

The truth value of the statement is True( T ) .