1. ex) I like her very much.

In this sentence, 'like and her' link together. They are pronounced as a linking sound or liaison just like one word. The h sound is omitted when they pronounced together.

2. Paying attention to the linking mark, listen carefully.
3. Listen carefully, and mark linking on the place where two words link together, pronouncing just like one word. ex) It's social science.
'It's social' is pronounced as a linking sound or liaison. The s sound in 'It's' is omitted, so you should pronounce just one 's' sound in 'It's social'.
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Are the expressions above grammatical? Would you check them?
I'd like to use correct expressions.

I agree with the grammar of the statements, but not what is said about dropping consonants and slurring words together.

Dropping "h's" was one of the things Eliza (in My Fair Lady) had to be taught NOT to do. It's ok to link the s sounds in "It's social" together , but saying "I liker" instead of "I like her" is sloppy speech.

1. The expression you provided, "I like her very much," is grammatically correct. However, the explanation you gave for the linking sound or liaison is not accurate. In this sentence, "like" and "her" do not actually link together or form a liaison. Each word is pronounced separately as "I like" and "her." There is no omission of the "h" sound.

2. The expression "Paying attention to the linking mark, listen carefully" is grammatically correct. However, it's not clear what you mean by "linking mark." If you are referring to the joining of words in pronunciation, it is more commonly known as liaison or linking.

3. The expression "Listen carefully, and mark linking on the place where two words link together, pronouncing just like one word. ex) It's social science" is grammatically correct, but the explanation of liaison is still inaccurate. In the example sentence, "It's social science," there is no specific liaison occurring. Each word is pronounced separately without any omission of sounds.

To improve the accuracy of your explanations, it would be better to provide clear and accurate examples of liaison or linking sounds, such as words like "an apple" pronounced as "an apple," where the "n" sound in "an" links with the beginning vowel sound in "apple."