5 FACTS ABOUT CROCODILES

1.They Have Ancient Lineage
Crocodiles have a remarkable evolutionary history and are often referred to as living fossils. They have remained relatively unchanged for over 200 million years, making them one of the oldest reptile lineages on Earth.

2.Their Size And Longevity Supersedes Most Species
Crocodiles are known for their impressive size. The largest species, the saltwater crocodile, can reach lengths of up to 20 feet (6 meters) and weigh over 2,000 pounds (907 kilograms). They also have a long lifespan, with some individuals living for more than 70 years.

3.They Have Incredibly Powerful Jaws
Crocodiles have one of the strongest bite forces in the animal kingdom. The muscles that close their jaws are incredibly powerful, allowing them to exert tremendous pressure when capturing prey. Their bite force is estimated to be several thousand pounds per square inch.

4.They Can Regenerate Their Teeth
Crocodiles have a unique tooth replacement system. They have a specialised groove in their jaws that allows new teeth to grow to replace the ones they lose. Over their lifetime, they can grow and replace thousands of teeth.

5.They Have A Sensorineural Organ
On the skin of a crocodile's snout, they have tiny pits called "integumentary sensory organs." These pits are sensitive to pressure changes in the water, allowing the crocodile to detect even the smallest disturbances made by potential prey or threats.

Closure Property
A non-empty set z is closed under a binary operation, * if for all a, b € Z.

Example: A binary operation * is defined on the set S= {0, 1, 2, 3, 4} by X*Y = x+y-xy. Find (a)2*4 (b)3*1 (c)0*3. Is the set S closed under the operation *?

Solution

(a)2*4, i.e, x = 2, y = 4

2+4 - (2x4) = 6-8 = -2

(b)3*1 = 3 + 1 - (3x1) = 4 - 3 =1

(c)0*3 = 0 + 3 - (0x3) = 3

Since -2€ S, therefore the operation * is not closed in S.

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Commutative Property
If set S, a non-empty set is closed under the binary operation *, for all a, b € S. Therefore, a binary operation is commutative if the order of combination does not affect the result.

Example; The operation * on the set R of real numbers is defined by p*q = p3 + q3 - 3qp.

Commutative condition p*q = q*p

To obtain q*p, use the same operation q*p, use the same operation p*q but replace p by q and q by p. Hence, q*p = p3 + q3 - 3qp

In conclusion p*q = q*p, the operation is commutative.

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Associative Property
If a non - empty set S is closed under a binary operation *, that is a*b € S. Then a binary operation is associative if (a*b) *c = a*(b*c)

Such that C also belongs to S.

Example: The operation Ө on the set Z of integers is defined by; a Ө b = 2a + 3b - 1. Determine whether or not the operation is associative in Z.

Solution

Introduce another element C

(aӨb)Өc = (2a+3b-1) Ө C

= 2(2a+3b-1) + 3c - 1

= 4a + 6b - 2 + 3c - 1

= 4a + 6b + 3c - 3

Also, the RHS, a Ө (b Ө c) = a Ө (2b + 3c - 1)

= 2a + 3(2b + 3c - 1) - 1

= 2a + 6b + 9c - 3 -1

a Ө (b Ө c) = 2a + 6b + 9c -4

Since, (a Ө b) Ө c ≠ a Ө (b Ө c), the operation is not associative in Z.

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Distributive Property
If a set is closed under two or more binary operations (*Ө) for all a,b and c € S, such that:

a*(bӨc) = (a*b)Ө(a*c) - Left distributive

(BӨc)*a = (b*a)Ө(c*a) - Right distributive over the operation Ө

Example: Given the set R of real numbers under the operations * and Ө defined by: a*b = a + b - 3, aӨb = 5ab for all a, b € R. Does * distribute over Ө.

Solution:

Let a, b, c € R

a * (bӨc) = (a*b) Ө (a*c)

a * (bӨc) = a*(5ab)

= a + 5ab - 3

(a*b) Ө (a*c) = (a+b-3) Ө (a+c-3)

=5(a+b-3)(a+c-3)

From the expansion, it's obvious that a*(bӨc) ≠ (a*b) Ө (a*c) therefore * does not distribute over

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