Rewrite each of the following statements in the form "∀ x, ." (a) All dinosaurs are extinct. ∀ x, . (b) Every real number is positive, negative, or zero. ∀ x, . (c) No irrational numbers are integers. ∀ x, .(d) No logicians are lazy. ∀ x, .(e) The number 2,147,581,953 is not equal to the square of any integer. ∀ x, .(f) The number −1 is not equal to the square of any real number.

See below

Step-by-step explanation:

Essentially, we have to replace "quantifier words" like "All", "Every" by the universal quantifier ∀.

a) ∀ dinosaur x, x is extinct.

b) ∀ real number x, x is positive, negative, or zero.

c) ∀ irrational number x, x is not an integer.

d) ∀ logician x, x is not lazy.

e) ∀ integer x, x²≠ 2,147,581,953.

f)  ∀ real number x, x²≠ -1.

In a) and b) we replace the words without major changes. In the other statements, we modify the statement using negation. For example, "No irrational numbers are integers." is equivalent to "Every irrational number is not integer".

Step-by-step explanation:

From the information given:

For 5 cards to be drawn where the first two are 2 spades and 6th of the spade.

Then the cards should be: 2,3,4,5,6  of spade.

Since 2 and 6 are already drawn.

Then;

the 3rd card maybe 3, 4, or 5 of the spade.

Thus, the probability that it is the third card is: 3/50

The probability is the 4th card 2/49 ;   &

The probability that it is the fifth card is 1/48

Thus, the probability that a 5-card straight flush is drawn is: